* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Logic and Proof - Numeracy Workshop
Fuzzy logic wikipedia , lookup
Peano axioms wikipedia , lookup
Willard Van Orman Quine wikipedia , lookup
Jesús Mosterín wikipedia , lookup
Turing's proof wikipedia , lookup
Quantum logic wikipedia , lookup
Gödel's incompleteness theorems wikipedia , lookup
Truth-bearer wikipedia , lookup
Combinatory logic wikipedia , lookup
Propositional calculus wikipedia , lookup
History of logic wikipedia , lookup
Modal logic wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Mathematical logic wikipedia , lookup
Laws of Form wikipedia , lookup
Natural deduction wikipedia , lookup
Intuitionistic logic wikipedia , lookup
Law of thought wikipedia , lookup
Please Note These pdf slides are configured for viewing on a computer screen. Viewing them on hand-held devices may be difficult as they require a “slideshow” mode. Do not try to print them out as there are many more pages than the number of slides listed at the bottom right of each screen. Apologies for any inconvenience. Logic and Proof Numeracy Workshop Adrian Dudek, Geoff Coates Adrian Dudek, Geoff Coates Logic and Proof 2 / 33 Introduction These slides give a brief introduction to mathematical logic and methods of proof Adrian Dudek, Geoff Coates Logic and Proof 3 / 33 Introduction These slides give a brief introduction to mathematical logic and methods of proof Workshop resources: These slides are available online: www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources Adrian Dudek, Geoff Coates Logic and Proof 3 / 33 Introduction These slides give a brief introduction to mathematical logic and methods of proof Workshop resources: These slides are available online: www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources Next Workshop: See your Workshop Calendar → www.studysmarter.uwa.edu.au Adrian Dudek, Geoff Coates Logic and Proof 3 / 33 Introduction These slides give a brief introduction to mathematical logic and methods of proof Workshop resources: These slides are available online: www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources Next Workshop: See your Workshop Calendar → www.studysmarter.uwa.edu.au Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202, Second Floor, Social Sciences South Building, every week. Adrian Dudek, Geoff Coates Logic and Proof 3 / 33 Introduction These slides give a brief introduction to mathematical logic and methods of proof Workshop resources: These slides are available online: www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources Next Workshop: See your Workshop Calendar → www.studysmarter.uwa.edu.au Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202, Second Floor, Social Sciences South Building, every week. Email: [email protected] Adrian Dudek, Geoff Coates Logic and Proof 3 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. The square root of two is irrational. Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. The square root of two is irrational. Squaring a number always makes it larger. Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. The square root of two is irrational. Squaring a number always makes it larger. Which of the above are true? Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. X The square root of two is irrational. Squaring a number always makes it larger. Which of the above are true? Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. X The square root of two is irrational. X Squaring a number always makes it larger. Which of the above are true? Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements Here are some examples of mathematical statements which are written in English: Adding one to a number always makes the number greater. X The square root of two is irrational. X Squaring a number always makes it larger. × Which of the above are true? Adrian Dudek, Geoff Coates Logic and Proof 4 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R, x + 1 > x. Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R, x + 1 > x. The square root of two is irrational: Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R, x + 1 > x. √ The square root of two is irrational: 2 ∈ Q Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R, x + 1 > x. √ √ / Q). The square root of two is irrational: 2 ∈ Q (or 2 ∈ Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R, x + 1 > x. √ √ / Q). The square root of two is irrational: 2 ∈ Q (or 2 ∈ Squaring a number always makes it larger: ∀x ∈ R Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Mathematical Statements The writing and reading maths workshop covered some shorthand notation to replace common mathematical words and phrases. Here is another one: “∀” stands for “for all” or “for every possible value of”. We can use this to shorten written statements: Adding one to a number always makes the number greater: ∀x ∈ R, x + 1 > x. √ √ / Q). The square root of two is irrational: 2 ∈ Q (or 2 ∈ Squaring a number always makes it larger: ∀x ∈ R, x 2 > x. Adrian Dudek, Geoff Coates Logic and Proof 5 / 33 Compound Statements and Connectives Statements can be combined, using logical connectives, to form compound statements. Adrian Dudek, Geoff Coates Logic and Proof 6 / 33 Compound Statements and Connectives Statements can be combined, using logical connectives, to form compound statements. Here we let A and B represent statements. Adrian Dudek, Geoff Coates Logic and Proof 6 / 33 Compound Statements and Connectives Statements can be combined, using logical connectives, to form compound statements. Here we let A and B represent statements. “and” Adrian Dudek, Geoff Coates Logic and Proof 6 / 33 Compound Statements and Connectives Statements can be combined, using logical connectives, to form compound statements. Here we let A and B represent statements. “and” The compound statement “A and B” (denoted A ∧ B) is true if A is true and B is true. Adrian Dudek, Geoff Coates Logic and Proof 6 / 33 Compound Statements and Connectives Statements can be combined, using logical connectives, to form compound statements. Here we let A and B represent statements. “and” The compound statement “A and B” (denoted A ∧ B) is true if A is true and B is true. The statement “It is raining and my socks are wet” is only true if both statements “it is raining” and “my socks are wet” are true. Adrian Dudek, Geoff Coates Logic and Proof 6 / 33 Compound Statements and Connectives “or” Adrian Dudek, Geoff Coates Logic and Proof 7 / 33 Compound Statements and Connectives “or” The compound statement “A or B” (denoted A ∨ B) is true if at least one of A or B is true. Adrian Dudek, Geoff Coates Logic and Proof 7 / 33 Compound Statements and Connectives “or” The compound statement “A or B” (denoted A ∨ B) is true if at least one of A or B is true. The statement “I want to go to the movies or I want to go to the party” is true if one of the following holds: Adrian Dudek, Geoff Coates Logic and Proof 7 / 33 Compound Statements and Connectives “or” The compound statement “A or B” (denoted A ∨ B) is true if at least one of A or B is true. The statement “I want to go to the movies or I want to go to the party” is true if one of the following holds: I want to go to the movies Adrian Dudek, Geoff Coates Logic and Proof 7 / 33 Compound Statements and Connectives “or” The compound statement “A or B” (denoted A ∨ B) is true if at least one of A or B is true. The statement “I want to go to the movies or I want to go to the party” is true if one of the following holds: I want to go to the movies I want to go to the party Adrian Dudek, Geoff Coates Logic and Proof 7 / 33 Compound Statements and Connectives “or” The compound statement “A or B” (denoted A ∨ B) is true if at least one of A or B is true. The statement “I want to go to the movies or I want to go to the party” is true if one of the following holds: I want to go to the movies I want to go to the party I want to go the movies and I want to go to the party Adrian Dudek, Geoff Coates Logic and Proof 7 / 33 Compound Statements and Connectives “not” Adrian Dudek, Geoff Coates Logic and Proof 8 / 33 Compound Statements and Connectives “not” The statement “not A” (denoted ∼ A), called the negation of A, is true if A is false. Adrian Dudek, Geoff Coates Logic and Proof 8 / 33 Compound Statements and Connectives “not” The statement “not A” (denoted ∼ A), called the negation of A, is true if A is false. Let A be the statement “π equals 3”. Then ∼ A is the statement π 6= 3. This is true if the statement “π equals 3” is false. Adrian Dudek, Geoff Coates Logic and Proof 8 / 33 Compound Statements and Connectives “implication” Adrian Dudek, Geoff Coates Logic and Proof 9 / 33 Compound Statements and Connectives “implication” The statement “If A is true, then B is true” (denoted A ⇒ B) is a true statement except when A is true and B is false. Adrian Dudek, Geoff Coates Logic and Proof 9 / 33 Compound Statements and Connectives “implication” The statement “If A is true, then B is true” (denoted A ⇒ B) is a true statement except when A is true and B is false. Note: If A is false, B could still be true. Adrian Dudek, Geoff Coates Logic and Proof 9 / 33 Compound Statements and Connectives “implication” The statement “If A is true, then B is true” (denoted A ⇒ B) is a true statement except when A is true and B is false. Note: If A is false, B could still be true. Consider the statement “If I am rich, then I have at least $10”. This is of the form A ⇒ B, where A is the statement “I am rich” and B is the statement “I have at least $10”. Adrian Dudek, Geoff Coates Logic and Proof 9 / 33 Compound Statements and Connectives “implication” The statement “If A is true, then B is true” (denoted A ⇒ B) is a true statement except when A is true and B is false. Note: If A is false, B could still be true. Consider the statement “If I am rich, then I have at least $10”. This is of the form A ⇒ B, where A is the statement “I am rich” and B is the statement “I have at least $10”. Note: If A ⇒ B is true, the reverse implication B ⇒ A may not be. Adrian Dudek, Geoff Coates Logic and Proof 9 / 33 Compound Statements and Connectives “implication” The statement “If A is true, then B is true” (denoted A ⇒ B) is a true statement except when A is true and B is false. Note: If A is false, B could still be true. Consider the statement “If I am rich, then I have at least $10”. This is of the form A ⇒ B, where A is the statement “I am rich” and B is the statement “I have at least $10”. Note: If A ⇒ B is true, the reverse implication B ⇒ A may not be. I know lots of people that have at least $10 but are not rich! Adrian Dudek, Geoff Coates Logic and Proof 9 / 33 Compound Statements and Connectives “double implication” Adrian Dudek, Geoff Coates Logic and Proof 10 / 33 Compound Statements and Connectives “double implication” The statement “A is true if and only if B is true” (denoted A ⇐⇒ B) means A ⇒ B and B ⇒ A. That is, the statements imply each other! Adrian Dudek, Geoff Coates Logic and Proof 10 / 33 Compound Statements and Connectives “double implication” The statement “A is true if and only if B is true” (denoted A ⇐⇒ B) means A ⇒ B and B ⇒ A. That is, the statements imply each other! Sometimes we say that A and B are equivalent. Adrian Dudek, Geoff Coates Logic and Proof 10 / 33 Compound Statements and Connectives “double implication” The statement “A is true if and only if B is true” (denoted A ⇐⇒ B) means A ⇒ B and B ⇒ A. That is, the statements imply each other! Sometimes we say that A and B are equivalent. Let A be the statement “x equals zero”, and let B be the statement “x 2 equals zero”. Adrian Dudek, Geoff Coates Logic and Proof 10 / 33 Compound Statements and Connectives “double implication” The statement “A is true if and only if B is true” (denoted A ⇐⇒ B) means A ⇒ B and B ⇒ A. That is, the statements imply each other! Sometimes we say that A and B are equivalent. Let A be the statement “x equals zero”, and let B be the statement “x 2 equals zero”. Then we have that A ⇐⇒ B as we have both A ⇒ B and B ⇒ A. Adrian Dudek, Geoff Coates Logic and Proof 10 / 33 Exercise Consider the following two statements: A : x2 = 9 B : x =3 Which of the following are true? A⇒B B⇒A A ⇐⇒ B Adrian Dudek, Geoff Coates Logic and Proof 11 / 33 Exercise Consider the following two statements: A : x2 = 9 B : x =3 Which of the following are true? A⇒B × B⇒AX A ⇐⇒ B × Adrian Dudek, Geoff Coates Logic and Proof 11 / 33 Exercise Consider the following two statements: A : x is an even number B : x + 2 is an even number Which of the following are true? A⇒B B⇒A A ⇐⇒ B Adrian Dudek, Geoff Coates Logic and Proof 12 / 33 Exercise Consider the following two statements: A : x is an even number B : x + 2 is an even number Which of the following are true? A⇒B X B⇒AX A ⇐⇒ B Adrian Dudek, Geoff Coates Logic and Proof 12 / 33 Exercise Consider the following two statements: A : x is an even number B : x + 2 is an even number Which of the following are true? A⇒B X B⇒AX A ⇐⇒ BX Adrian Dudek, Geoff Coates Logic and Proof 12 / 33 The Converse “converse” Adrian Dudek, Geoff Coates Logic and Proof 13 / 33 The Converse “converse” B ⇒ A is the converse of A ⇒ B. It’s as easy as reversing the direction of the arrow! Adrian Dudek, Geoff Coates Logic and Proof 13 / 33 The Converse “converse” B ⇒ A is the converse of A ⇒ B. It’s as easy as reversing the direction of the arrow! The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. Adrian Dudek, Geoff Coates Logic and Proof 13 / 33 The Converse “converse” B ⇒ A is the converse of A ⇒ B. It’s as easy as reversing the direction of the arrow! The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . Adrian Dudek, Geoff Coates Logic and Proof 13 / 33 The Converse “converse” B ⇒ A is the converse of A ⇒ B. It’s as easy as reversing the direction of the arrow! The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because Adrian Dudek, Geoff Coates Logic and Proof 13 / 33 The Converse “converse” B ⇒ A is the converse of A ⇒ B. It’s as easy as reversing the direction of the arrow! The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because x could be equal to −2. Adrian Dudek, Geoff Coates Logic and Proof 13 / 33 The Contrapositive “contrapositive” Adrian Dudek, Geoff Coates Logic and Proof 14 / 33 The Contrapositive “contrapositive” ∼ B ⇒∼ A is the contrapositive of A ⇒ B. Adrian Dudek, Geoff Coates Logic and Proof 14 / 33 The Contrapositive “contrapositive” ∼ B ⇒∼ A is the contrapositive of A ⇒ B. A statement and its contrapositive are logically equivalent. This means that if one is true, then the other is true! Adrian Dudek, Geoff Coates Logic and Proof 14 / 33 The Contrapositive “contrapositive” ∼ B ⇒∼ A is the contrapositive of A ⇒ B. A statement and its contrapositive are logically equivalent. This means that if one is true, then the other is true! For example, x = 2 ⇒ x2 = 4 is true, and its contrapositive Adrian Dudek, Geoff Coates Logic and Proof 14 / 33 The Contrapositive “contrapositive” ∼ B ⇒∼ A is the contrapositive of A ⇒ B. A statement and its contrapositive are logically equivalent. This means that if one is true, then the other is true! For example, x = 2 ⇒ x2 = 4 is true, and its contrapositive x 2 6= 4 ⇒ Adrian Dudek, Geoff Coates Logic and Proof 14 / 33 The Contrapositive “contrapositive” ∼ B ⇒∼ A is the contrapositive of A ⇒ B. A statement and its contrapositive are logically equivalent. This means that if one is true, then the other is true! For example, x = 2 ⇒ x2 = 4 is true, and its contrapositive x 2 6= 4 ⇒ x 6= 2 is true. Adrian Dudek, Geoff Coates Logic and Proof 14 / 33 The Contrapositive: Example Consider the statement: x ≥ 2 ⇒ x2 > 1 The contrapositive of the above is the statement: Adrian Dudek, Geoff Coates Logic and Proof 15 / 33 The Contrapositive: Example Consider the statement: x ≥ 2 ⇒ x2 > 1 The contrapositive of the above is the statement: x2 ≤ 1 Adrian Dudek, Geoff Coates Logic and Proof 15 / 33 The Contrapositive: Example Consider the statement: x ≥ 2 ⇒ x2 > 1 The contrapositive of the above is the statement: x2 ≤ 1 ⇒ x < 2 Both are true. Adrian Dudek, Geoff Coates Logic and Proof 15 / 33 Necessary and Sufficient Conditions If the statement A ⇒ B is true then we say “A is a sufficient condition for B”, i.e. the truth of A is sufficient to guarantee that B is true. Adrian Dudek, Geoff Coates Logic and Proof 16 / 33 Necessary and Sufficient Conditions If the statement A ⇒ B is true then we say “A is a sufficient condition for B”, i.e. the truth of A is sufficient to guarantee that B is true. Example: “I am rich” ⇒ “I have at least $10” so if “I am rich” (A) is true, it guarantees that “I have at least $10” (B). Adrian Dudek, Geoff Coates Logic and Proof 16 / 33 Necessary and Sufficient Conditions If the statement A ⇒ B is true then we say “A is a sufficient condition for B”, i.e. the truth of A is sufficient to guarantee that B is true. Example: “I am rich” ⇒ “I have at least $10” so if “I am rich” (A) is true, it guarantees that “I have at least $10” (B). We also say that “B is a necessary condition for A”, i.e., if B is false then A is false. Adrian Dudek, Geoff Coates Logic and Proof 16 / 33 Necessary and Sufficient Conditions If the statement A ⇒ B is true then we say “A is a sufficient condition for B”, i.e. the truth of A is sufficient to guarantee that B is true. Example: “I am rich” ⇒ “I have at least $10” so if “I am rich” (A) is true, it guarantees that “I have at least $10” (B). We also say that “B is a necessary condition for A”, i.e., if B is false then A is false. Example: If “I have at least $10” (B) is false, then I can’t be rich (A). Adrian Dudek, Geoff Coates Logic and Proof 16 / 33 Necessary and Sufficient Conditions If the statement A ⇒ B is true then we say “A is a sufficient condition for B”, i.e. the truth of A is sufficient to guarantee that B is true. Example: “I am rich” ⇒ “I have at least $10” so if “I am rich” (A) is true, it guarantees that “I have at least $10” (B). We also say that “B is a necessary condition for A”, i.e., if B is false then A is false. Example: If “I have at least $10” (B) is false, then I can’t be rich (A). If the statement A ⇐⇒ B is true, then we say that “A is necessary and sufficient for B”. Adrian Dudek, Geoff Coates Logic and Proof 16 / 33 Proof Some might say that mathematics is a “toolkit of truths”. Adrian Dudek, Geoff Coates Logic and Proof 17 / 33 Proof Some might say that mathematics is a “toolkit of truths”. Research in mathematics consists of increasing the number of tools in our toolkit. Adrian Dudek, Geoff Coates Logic and Proof 17 / 33 Proof Some might say that mathematics is a “toolkit of truths”. Research in mathematics consists of increasing the number of tools in our toolkit. The way that we prove new tools, is to show that they are implications of tools which are already in our toolkit. Adrian Dudek, Geoff Coates Logic and Proof 17 / 33 Proof Some might say that mathematics is a “toolkit of truths”. Research in mathematics consists of increasing the number of tools in our toolkit. The way that we prove new tools, is to show that they are implications of tools which are already in our toolkit. There are two main kinds of tool: axioms and theorems. Adrian Dudek, Geoff Coates Logic and Proof 17 / 33 Axioms Axioms are statements that are simply accepted as being true without the need for proof. This is because they are so fundamental that we feel everyone must accept them. Adrian Dudek, Geoff Coates Logic and Proof 18 / 33 Axioms Axioms are statements that are simply accepted as being true without the need for proof. This is because they are so fundamental that we feel everyone must accept them. Here are some examples: Adrian Dudek, Geoff Coates Logic and Proof 18 / 33 Axioms Axioms are statements that are simply accepted as being true without the need for proof. This is because they are so fundamental that we feel everyone must accept them. Here are some examples: Euclid’s First Axiom: There is only one straight line that can be drawn to join two specific points in space. Adrian Dudek, Geoff Coates Logic and Proof 18 / 33 Axioms Axioms are statements that are simply accepted as being true without the need for proof. This is because they are so fundamental that we feel everyone must accept them. Here are some examples: Euclid’s First Axiom: There is only one straight line that can be drawn to join two specific points in space. Well Ordering Principle: Every non-empty set of positive integers contains a smallest element. Adrian Dudek, Geoff Coates Logic and Proof 18 / 33 Theorems Theorems are statements that can be proved to be true using accepted definitions, axioms, and other already proven theorems. Adrian Dudek, Geoff Coates Logic and Proof 19 / 33 Theorems Theorems are statements that can be proved to be true using accepted definitions, axioms, and other already proven theorems. The truth of the theorem is arrived at by reasoning from other accepted truths. Adrian Dudek, Geoff Coates Logic and Proof 19 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 8 × 2 = 16 Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 8 × 2 = 16 4 × 4 = 16 Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 8 × 2 = 16 4 × 4 = 16 40 × 8 = 320 Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 8 × 2 = 16 4 × 4 = 16 40 × 8 = 320 We always seem to get an even number. Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 8 × 2 = 16 4 × 4 = 16 40 × 8 = 320 We always seem to get an even number. However, we can not conclude that the statement “the product of any two even numbers is even” is a true statement, simply because we have seen a few examples. Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra Look what happens when we multiply two even numbers together: 4 × 6 = 24 8 × 2 = 16 4 × 4 = 16 40 × 8 = 320 We always seem to get an even number. However, we can not conclude that the statement “the product of any two even numbers is even” is a true statement, simply because we have seen a few examples. We need to prove this theorem completely. This means that we have to present a convincing argument. Adrian Dudek, Geoff Coates Logic and Proof 20 / 33 Deductive Proofs using Algebra We want to show that for any two even numbers we choose, their product is also even. Adrian Dudek, Geoff Coates Logic and Proof 21 / 33 Deductive Proofs using Algebra We want to show that for any two even numbers we choose, their product is also even. However, we don’t want to write out every single choice of two even numbers as this is impossible! Adrian Dudek, Geoff Coates Logic and Proof 21 / 33 Deductive Proofs using Algebra We want to show that for any two even numbers we choose, their product is also even. However, we don’t want to write out every single choice of two even numbers as this is impossible! Instead, we use expressions which represent arbitrary even numbers. Adrian Dudek, Geoff Coates Logic and Proof 21 / 33 Deductive Proofs using Algebra We want to show that for any two even numbers we choose, their product is also even. However, we don’t want to write out every single choice of two even numbers as this is impossible! Instead, we use expressions which represent arbitrary even numbers. By definition, even numbers are multiples of two, that is, any even number is the product of 2 and some other integer. Adrian Dudek, Geoff Coates Logic and Proof 21 / 33 Deductive Proofs using Algebra We want to show that for any two even numbers we choose, their product is also even. However, we don’t want to write out every single choice of two even numbers as this is impossible! Instead, we use expressions which represent arbitrary even numbers. By definition, even numbers are multiples of two, that is, any even number is the product of 2 and some other integer. Hence, we can represent an even number by the expression 2n, where n is an integer. Adrian Dudek, Geoff Coates Logic and Proof 21 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Multiply them together: (2n)(2m) Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Multiply them together: (2n)(2m) We can simplify and express the above product in a clever way! (2n)(2m) = 4nm Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Multiply them together: (2n)(2m) We can simplify and express the above product in a clever way! (2n)(2m) = 4nm = 2(2nm) Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Multiply them together: (2n)(2m) We can simplify and express the above product in a clever way! (2n)(2m) = 4nm = 2(2nm) But 2(2nm) is really just the product of 2 and an integer (ie. 2nm)! So this product is clearly even! Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Multiply them together: (2n)(2m) We can simplify and express the above product in a clever way! (2n)(2m) = 4nm = 2(2nm) But 2(2nm) is really just the product of 2 and an integer (ie. 2nm)! So this product is clearly even! Note: We usually denote the end of a proof with a box. Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: even x even = even Here is our proof. Take two even numbers: 2n, 2m Multiply them together: (2n)(2m) We can simplify and express the above product in a clever way! (2n)(2m) = 4nm = 2(2nm) But 2(2nm) is really just the product of 2 and an integer (ie. 2nm)! So this product is clearly even! Note: We usually denote the end of a proof with a box. Adrian Dudek, Geoff Coates Logic and Proof 22 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Take two odd numbers: 2n + 1, 2m + 1 Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Take two odd numbers: 2n + 1, 2m + 1 Add them together: 2n + 1 + 2m + 1 Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Take two odd numbers: 2n + 1, 2m + 1 Add them together: 2n + 1 + 2m + 1 We can simplify and express the above in the same clever way: 2n + 1 + 2m + 1 = 2n + 2m + 2 Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Take two odd numbers: 2n + 1, 2m + 1 Add them together: 2n + 1 + 2m + 1 We can simplify and express the above in the same clever way: 2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1) Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Take two odd numbers: 2n + 1, 2m + 1 Add them together: 2n + 1 + 2m + 1 We can simplify and express the above in the same clever way: 2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1) We have managed to write the sum of two odd numbers as a number which is a multiple of 2! So the sum of two odd numbers must be an even number. Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Proof: odd + odd = even Let’s prove that the sum of two odd numbers is even. Here is our proof. Take two odd numbers: 2n + 1, 2m + 1 Add them together: 2n + 1 + 2m + 1 We can simplify and express the above in the same clever way: 2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1) We have managed to write the sum of two odd numbers as a number which is a multiple of 2! So the sum of two odd numbers must be an even number. Adrian Dudek, Geoff Coates Logic and Proof 23 / 33 Structure of algebraic deductive proof Theorem: If statement(s) is/are true, show that statement is true. Adrian Dudek, Geoff Coates Logic and Proof 24 / 33 Structure of algebraic deductive proof Theorem: If statement(s) is/are true, show that statement is true. Proof: statement(s) given Structure of algebraic deductive proof Theorem: If statement(s) is/are true, show that statement is true. Proof: statement(s) given statement to be proved Structure of algebraic deductive proof Theorem: If statement(s) is/are true, show that statement is true. Proof: statement(s) given clever, legal steps statement to be proved Structure of algebraic deductive proof Theorem: If statement(s) is/are true, show that statement is true. Proof: statement(s) given clever, legal steps clever, legal steps statement to be proved Adrian Dudek, Geoff Coates Logic and Proof 24 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. A−1 Adrian Dudek, Geoff Coates T Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: A−1 Adrian Dudek, Geoff Coates T Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A−1 Adrian Dudek, Geoff Coates T Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric A−1 Adrian Dudek, Geoff Coates T Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible A−1 Adrian Dudek, Geoff Coates T Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible A−1 is symmetric A −1 T Adrian Dudek, Geoff Coates Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible clever, legal steps A−1 is symmetric Proofs are hard because “clever” steps may be hard to spot. Here are some tips. A−1 Adrian Dudek, Geoff Coates Logic and Proof T 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible A−1 is symmetric T T A−1 Tip: Express the red and blue phrases algebraically. A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A A−1 is symmetric T T A−1 Tip: Express the red and blue phrases algebraically. A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and A−1 exists A−1 is symmetric T T A−1 Tip: Express the red and blue phrases algebraically. A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and A−1 exists A−1 T = A−1 A−1 is symmetric T T A−1 Tip: Express the red and blue phrases algebraically. A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and A−1 exists A−1 T = A−1 A−1 is symmetric T T A−1 We need to start with an equation involving A−1 . . . A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ A−1 exists AA−1 = I A−1 T (from defn of inverse) = A−1 A−1 is symmetric T T A−1 We need to start with an equation involving A−1 . . . A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ A−1 exists AA−1 = I A−1 T (from defn of inverse) = A−1 A−1 is symmetric T T A−1 Now to get transpose involved . . . A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ ⇒ A−1 exists AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) A−1 T = A−1 A−1 is symmetric T T A−1 Now to get transpose involved . . . A−1 Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ ⇒ A−1 exists AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) A−1 T = A−1 A−1 is symmetric T We need to get an A−1 term. Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ ⇒ ⇒ A−1 exists AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) T A−1 AT = I (a property of transpose) A−1 T = A−1 A−1 is symmetric T We need to get an A−1 term. Adrian Dudek, Geoff Coates Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ ⇒ ⇒ ⇒ A−1 exists AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) T A−1 AT = I (a property of transpose) T A−1 A = I (known property of A) A−1 T = A−1 A−1 is symmetric A −1 T Adrian Dudek, Geoff Coates Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ ⇒ ⇒ ⇒ ⇒ A−1 exists AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) T A−1 AT = I (a property of transpose) T A−1 A = I (known property of A) T A−1 is the inverse of A by defn of inverse T A−1 = A−1 A−1 is symmetric A −1 T Adrian Dudek, Geoff Coates Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ A−1 exists AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) T A−1 AT = I (a property of transpose) T A−1 A = I (known property of A) T A−1 is the inverse of A by defn of inverse T A−1 = A−1 A−1 is symmetric A −1 T Adrian Dudek, Geoff Coates Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) T A−1 AT = I (a property of transpose) T A−1 A = I (known property of A) T A−1 is the inverse of A by defn of inverse T A−1 = A−1 ∴ A−1 is symmetric ⇒ ⇒ ⇒ ⇒ ⇒ A −1 T Adrian Dudek, Geoff Coates A−1 exists Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Structure of algebraic deductive proof: example (This example involves matrices and is similar to proofs in MATH1001.) Theorem: If A is a symmetric, invertible matrix then A−1 is also symmetric. Proof: We know A is symmetric and A is invertible AT = A and ⇒ AA−1 = I (from defn of inverse) T AA−1 = (I )T = I (transpose both sides) T A−1 AT = I (a property of transpose) T A−1 A = I (known property of A) T A−1 is the inverse of A by defn of inverse T A−1 = A−1 ∴ A−1 is symmetric ⇒ ⇒ ⇒ ⇒ ⇒ A −1 T Adrian Dudek, Geoff Coates A−1 exists Express the red and blue phrases algebraically. Logic and Proof 25 / 33 Proof or Counterexample To prove that a statement is true, we need to construct a proof. To show that a statement is false, we simply need to find a single counterexample where it fails. Adrian Dudek, Geoff Coates Logic and Proof 26 / 33 Proof or Counterexample: Example Consider the following statement: Squaring a number makes it larger. Adrian Dudek, Geoff Coates Logic and Proof 27 / 33 Proof or Counterexample: Example Consider the following statement: Squaring a number makes it larger. This statement is false. To disprove it, we simply need to demonstrate one instance where it fails to be true. Adrian Dudek, Geoff Coates Logic and Proof 27 / 33 Proof or Counterexample: Example Consider the following statement: Squaring a number makes it larger. This statement is false. To disprove it, we simply need to demonstrate one instance where it fails to be true. We can see that if we square the number 1, we get 1, which is not larger. Adrian Dudek, Geoff Coates Logic and Proof 27 / 33 Proof or Counterexample: Example Consider the following statement: Squaring a number makes it larger. This statement is false. To disprove it, we simply need to demonstrate one instance where it fails to be true. We can see that if we square the number 1, we get 1, which is not larger. So the above statement can not be true. Adrian Dudek, Geoff Coates Logic and Proof 27 / 33 Types of Proof: Contrapositives There are many common types of proof. We have already seen deductive proofs using algebra. Adrian Dudek, Geoff Coates Logic and Proof 28 / 33 Types of Proof: Contrapositives There are many common types of proof. We have already seen deductive proofs using algebra. We know that a statement is true, if and only if its contrapositive is true. Sometimes it turns out that it’s easier to prove the contrapositive of a statement rather than the original statement itself. This is useful! Adrian Dudek, Geoff Coates Logic and Proof 28 / 33 Types of Proof: Proof by Contradiction Proof by Contradiction works as follows. Adrian Dudek, Geoff Coates Logic and Proof 29 / 33 Types of Proof: Proof by Contradiction Proof by Contradiction works as follows. 1 We are asked to prove that a statement is true. Adrian Dudek, Geoff Coates Logic and Proof 29 / 33 Types of Proof: Proof by Contradiction Proof by Contradiction works as follows. 1 We are asked to prove that a statement is true. 2 We instead assume that the statement is false. Adrian Dudek, Geoff Coates Logic and Proof 29 / 33 Types of Proof: Proof by Contradiction Proof by Contradiction works as follows. 1 We are asked to prove that a statement is true. 2 We instead assume that the statement is false. 3 We then show that this assumption leads to a contradiction. Adrian Dudek, Geoff Coates Logic and Proof 29 / 33 Types of Proof: Proof by Contradiction Proof by Contradiction works as follows. 1 We are asked to prove that a statement is true. 2 We instead assume that the statement is false. 3 We then show that this assumption leads to a contradiction. 4 Reality forces the original statement to be true! Adrian Dudek, Geoff Coates Logic and Proof 29 / 33 Proof by Contradiction: Example Use Proof by Contradiction to prove that Adrian Dudek, Geoff Coates Logic and Proof √ 2 is irrational. (Euclid 500BC) 30 / 33 Proof by Contradiction: Example Use Proof by Contradiction to prove that √ 2 is irrational. (Euclid 500BC) √ √ a 2 is rational, that is, 2 = where a and b are integers with no b common factors. (If they did have common factors we could cancel them out.) Proof: Assume that Adrian Dudek, Geoff Coates Logic and Proof 30 / 33 Proof by Contradiction: Example Use Proof by Contradiction to prove that √ 2 is irrational. (Euclid 500BC) √ √ a 2 is rational, that is, 2 = where a and b are integers with no b common factors. (If they did have common factors we could cancel them out.) Proof: Assume that We want to show that this assumption yields a contradiction. Adrian Dudek, Geoff Coates Logic and Proof 30 / 33 Proof by Contradiction: Example Use Proof by Contradiction to prove that √ 2 is irrational. (Euclid 500BC) √ √ a 2 is rational, that is, 2 = where a and b are integers with no b common factors. (If they did have common factors we could cancel them out.) Proof: Assume that We want to show that this assumption yields a contradiction. Squaring both sides of 2= Adrian Dudek, Geoff Coates √ a 2 = gives us: b a2 b2 Logic and Proof 30 / 33 Proof by Contradiction: Example Use Proof by Contradiction to prove that √ 2 is irrational. (Euclid 500BC) √ √ a 2 is rational, that is, 2 = where a and b are integers with no b common factors. (If they did have common factors we could cancel them out.) Proof: Assume that We want to show that this assumption yields a contradiction. Squaring both sides of 2= √ a 2 = gives us: b a2 b2 Rearranging this we get: 2b 2 = a2 Adrian Dudek, Geoff Coates Logic and Proof 30 / 33 Proof by Contradiction: Example 2b 2 = a2 Adrian Dudek, Geoff Coates Logic and Proof 31 / 33 Proof by Contradiction: Example 2b 2 = a2 We see that a2 must be even, because it is the product of 2 and some integer b 2 . Adrian Dudek, Geoff Coates Logic and Proof 31 / 33 Proof by Contradiction: Example 2b 2 = a2 We see that a2 must be even, because it is the product of 2 and some integer b 2 . Now, we proved earlier (ie. added to our maths toolkit) that the product of two even numbers is even. It’s not hard to show that the product of two odd numbers is odd (exercise) so, if a2 is even, then a must also be even. Adrian Dudek, Geoff Coates Logic and Proof 31 / 33 Proof by Contradiction: Example 2b 2 = a2 We see that a2 must be even, because it is the product of 2 and some integer b 2 . Now, we proved earlier (ie. added to our maths toolkit) that the product of two even numbers is even. It’s not hard to show that the product of two odd numbers is odd (exercise) so, if a2 is even, then a must also be even. As a is even, we can write a = 2n for some integer n. Substituting this in we get: 2b 2 = (2n)2 Adrian Dudek, Geoff Coates Logic and Proof 31 / 33 Proof by Contradiction: Example 2b 2 = a2 We see that a2 must be even, because it is the product of 2 and some integer b 2 . Now, we proved earlier (ie. added to our maths toolkit) that the product of two even numbers is even. It’s not hard to show that the product of two odd numbers is odd (exercise) so, if a2 is even, then a must also be even. As a is even, we can write a = 2n for some integer n. Substituting this in we get: 2b 2 = (2n)2 Expanding the bracket we get: 2b 2 = 4n2 Adrian Dudek, Geoff Coates Logic and Proof 31 / 33 Proof by Contradiction: Example 2b 2 = a2 We see that a2 must be even, because it is the product of 2 and some integer b 2 . Now, we proved earlier (ie. added to our maths toolkit) that the product of two even numbers is even. It’s not hard to show that the product of two odd numbers is odd (exercise) so, if a2 is even, then a must also be even. As a is even, we can write a = 2n for some integer n. Substituting this in we get: 2b 2 = (2n)2 Expanding the bracket we get: 2b 2 = 4n2 Dividing by 2 we get: b 2 = 2n2 Adrian Dudek, Geoff Coates Logic and Proof 31 / 33 Proof by Contradiction: Example b 2 = 2n2 Adrian Dudek, Geoff Coates Logic and Proof 32 / 33 Proof by Contradiction: Example b 2 = 2n2 Here we can see that b 2 must also be even, as it is the product of 2 and some other integer n2 . So b must also be even. Adrian Dudek, Geoff Coates Logic and Proof 32 / 33 Proof by Contradiction: Example b 2 = 2n2 Here we can see that b 2 must also be even, as it is the product of 2 and some other integer n2 . So b must also be even. We have established that both a and b are even. This means they both are divisible by 2. Adrian Dudek, Geoff Coates Logic and Proof 32 / 33 Proof by Contradiction: Example b 2 = 2n2 Here we can see that b 2 must also be even, as it is the product of 2 and some other integer n2 . So b must also be even. We have established that both a and b are even. This means they both are divisible by 2. However, we assumed that they had no factors in common. Thus, we have reached a contradiction. Adrian Dudek, Geoff Coates Logic and Proof 32 / 33 Proof by Contradiction: Example b 2 = 2n2 Here we can see that b 2 must also be even, as it is the product of 2 and some other integer n2 . So b must also be even. We have established that both a and b are even. This means they both are divisible by 2. However, we assumed that they had no factors in common. Thus, we have reached a contradiction. ∴ Adrian Dudek, Geoff Coates √ 2 is irrational. Logic and Proof 32 / 33 Using STUDYSmarter Resources This resource was developed for UWA students by the STUDYSmarter team for the numeracy program. When using our resources, please retain them in their original form with both the STUDYSmarter heading and the UWA crest. Adrian Dudek, Geoff Coates Logic and Proof 33 / 33