Study Guide Unit Test2 with Sample Problems
... 3. Know how to prove statements using direct proofs and mathematical induction. 3.1. Prove that the sum of two odd numbers is an even number. Proof: Let x and y be two odd numbers. By the definition of odd numbers, there is some p integer such that x = 2p -1, and there is some q integer such that y ...
... 3. Know how to prove statements using direct proofs and mathematical induction. 3.1. Prove that the sum of two odd numbers is an even number. Proof: Let x and y be two odd numbers. By the definition of odd numbers, there is some p integer such that x = 2p -1, and there is some q integer such that y ...
How To Prove It
... It is based on the fact that the sum of the first n odd numbers is n2. For example, 1 + 3 = 4 = 22 and 1 + 3 + 5 = 9 = 32. So to show that (2n − 1)2 is 1 more than a multiple of 8, we must show that the sum of the first 2n − 1 odd numbers is 1 more than a multiple of 8. Now we observe the following ...
... It is based on the fact that the sum of the first n odd numbers is n2. For example, 1 + 3 = 4 = 22 and 1 + 3 + 5 = 9 = 32. So to show that (2n − 1)2 is 1 more than a multiple of 8, we must show that the sum of the first 2n − 1 odd numbers is 1 more than a multiple of 8. Now we observe the following ...
Hilbert`s Program Then and Now - Philsci
... logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematic ...
... logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematic ...
MATH/EECS 1019 Third test (version 1) – Fall 2014 Solutions 1. (3
... passes through exactly two of the 2m regions. Since it cuts each of these two regions in two parts, it creates a total of 2m + 2 regions. Hence by the principle of mathematical induction, the given statement is true. 7. (3 points) A function f (x) is said to be strictly increasing if f (b) > f (a) f ...
... passes through exactly two of the 2m regions. Since it cuts each of these two regions in two parts, it creates a total of 2m + 2 regions. Hence by the principle of mathematical induction, the given statement is true. 7. (3 points) A function f (x) is said to be strictly increasing if f (b) > f (a) f ...
Mathematical Statements and Their Proofs
... Let a and b be two even integers. Since they are even, they can be written in the forms a = 2x and b = 2y for some integers x and y, respectively. Then, a + b can be written in the form 2x + 2y, giving the following equation: a + b = 2x + 2y = 2(x + y) From this, we see that a + b is divisible by 2. ...
... Let a and b be two even integers. Since they are even, they can be written in the forms a = 2x and b = 2y for some integers x and y, respectively. Then, a + b can be written in the form 2x + 2y, giving the following equation: a + b = 2x + 2y = 2(x + y) From this, we see that a + b is divisible by 2. ...
A counterexample to the infinite version of a
... = {1 , 2, 3, ... , ) is divided in any manner into two sets ...
... = {1 , 2, 3, ... , ) is divided in any manner into two sets ...
Lecture 3. Mathematical Induction
... Induction plays a very important role in our knowledge about the world and nature. Practically all modern information (scientific and practical) is based on inductive reasoning. Past experience is used as the basis for generalizations about future experience. Induction in natural sciences cannot be ...
... Induction plays a very important role in our knowledge about the world and nature. Practically all modern information (scientific and practical) is based on inductive reasoning. Past experience is used as the basis for generalizations about future experience. Induction in natural sciences cannot be ...
Hilbert Calculus
... By induction on the structure of F (and using Lemma III): Atomic formulas: F = A. Easy. Negation: F = ¬G. We have: A(F ) = 1 iff A(G) = 0 iff G 6∈ S iff ¬G ∈ S iff F ∈ S. Implication: F = F1 → F2 . We have: A(F ) = 1 iff A(F1 → F2 ) = 1 iff (A(F1 ) = 0 or A(F2 ) = 1) iff (F1 6∈ S or F2 ∈ S) iff F1 → ...
... By induction on the structure of F (and using Lemma III): Atomic formulas: F = A. Easy. Negation: F = ¬G. We have: A(F ) = 1 iff A(G) = 0 iff G 6∈ S iff ¬G ∈ S iff F ∈ S. Implication: F = F1 → F2 . We have: A(F ) = 1 iff A(F1 → F2 ) = 1 iff (A(F1 ) = 0 or A(F2 ) = 1) iff (F1 6∈ S or F2 ∈ S) iff F1 → ...
Practice Questions
... Inductive Hypothesis: Every natural number less than n is a prime or a perfect square. Inductive step: Consider n. If n is prime, then we are done. Otherwise, n can be factored as n = rs with r and s less than or equal to n − 1. By the inductive hypothesis, r and s are perfect squares, so r = u2 and ...
... Inductive Hypothesis: Every natural number less than n is a prime or a perfect square. Inductive step: Consider n. If n is prime, then we are done. Otherwise, n can be factored as n = rs with r and s less than or equal to n − 1. By the inductive hypothesis, r and s are perfect squares, so r = u2 and ...
Characterizing integers among rational numbers
... Remark 4.2. Here we show that if f (t), g(t) ∈ Q(t) are rational functions, then the intersection of Tf (a),g(a) over all a ∈ Q such that f (a) and g(a) are nonzero and not both negative is always much larger than Z; this foils one possible approach to defining Z in Q using just one universal quant ...
... Remark 4.2. Here we show that if f (t), g(t) ∈ Q(t) are rational functions, then the intersection of Tf (a),g(a) over all a ∈ Q such that f (a) and g(a) are nonzero and not both negative is always much larger than Z; this foils one possible approach to defining Z in Q using just one universal quant ...
Logic (Mathematics 1BA1) Reminder: Sets of numbers Proof by
... Lets define p : x + y > 2 and q : (x > 1 ∨ y > 1), we can prove by contrapositive that p ⇒ q is true by showing that ¬q ⇒ ¬p is true. By contrapositive, this is equivalent2 to proving (x < 1 ∧ y < 1) ⇒ x + y < 2 ...
... Lets define p : x + y > 2 and q : (x > 1 ∨ y > 1), we can prove by contrapositive that p ⇒ q is true by showing that ¬q ⇒ ¬p is true. By contrapositive, this is equivalent2 to proving (x < 1 ∧ y < 1) ⇒ x + y < 2 ...
Part I - UMD Math - University of Maryland
... were two juniors and a number of seniors playing in a high school chess championship. The total score of the juniors was 8 points, and each senior got the same score. There were at most 12 seniors. How many seniors played in the tournament? a. 2 b. 6 c. 7 d. 10 e. 12 16. Let S be the set of all inte ...
... were two juniors and a number of seniors playing in a high school chess championship. The total score of the juniors was 8 points, and each senior got the same score. There were at most 12 seniors. How many seniors played in the tournament? a. 2 b. 6 c. 7 d. 10 e. 12 16. Let S be the set of all inte ...
Gödel on Conceptual Realism and Mathematical Intuition
... evidence of truth of propositions to be taken as axioms (i.e. as the basis of proof): intuition is a means of discovering new truths (they are somehow already implicit in our concepts, but not logically derivable from the axioms we have already accepted concerning these concepts). [9] Gaifman conten ...
... evidence of truth of propositions to be taken as axioms (i.e. as the basis of proof): intuition is a means of discovering new truths (they are somehow already implicit in our concepts, but not logically derivable from the axioms we have already accepted concerning these concepts). [9] Gaifman conten ...
Section 1.1: The irrationality of 2 . 1. This section introduces many of
... as the constant functions. So Q ⊂ R ⊂ R( x) . Since the irrationals in R are described as “filling in the gaps in Q,” how can there be even more elements in R(x)? What are these elements? What could go wrong with doing calculus on this even larger field? Note: Question 5 raises challenging issues th ...
... as the constant functions. So Q ⊂ R ⊂ R( x) . Since the irrationals in R are described as “filling in the gaps in Q,” how can there be even more elements in R(x)? What are these elements? What could go wrong with doing calculus on this even larger field? Note: Question 5 raises challenging issues th ...
Math Camp Notes: Basic Proof Techniques
... A, and we cannot prove it directly. However, we can A are impossible. Then we have indirectly proved that A must be true. Therefore, the we can prove A ⇒ B by rst assuming that A 6⇒ B and nding a contradiction. In other words, we start o by assuming that A is true but B is not. If this leads to a ...
... A, and we cannot prove it directly. However, we can A are impossible. Then we have indirectly proved that A must be true. Therefore, the we can prove A ⇒ B by rst assuming that A 6⇒ B and nding a contradiction. In other words, we start o by assuming that A is true but B is not. If this leads to a ...
REDUCTIO AD ABSURDUM* (Proof by contradiction) Y.K. Leong
... Let A 1 8 1 A 2 be a given triangle. On the line through A 1 A 2 produced, construct congruent triangles Ai8iAi+l• i = 2, 3, ... , n, where A 1 A 2 = A 2 A 3 = ... = AnAn+l· Let the angles of triangle A 1 8 1 A 2 be ex, {3, 'Y (see figure). We wish to show that ex + (3 + 'Y < 2 right angles. Suppos ...
... Let A 1 8 1 A 2 be a given triangle. On the line through A 1 A 2 produced, construct congruent triangles Ai8iAi+l• i = 2, 3, ... , n, where A 1 A 2 = A 2 A 3 = ... = AnAn+l· Let the angles of triangle A 1 8 1 A 2 be ex, {3, 'Y (see figure). We wish to show that ex + (3 + 'Y < 2 right angles. Suppos ...
Proofs • A theorem is a mathematical statement that can be shown to
... • A theorem is a mathematical statement that can be shown to be true. • An axiom or postulate is an assumption accepted without proof. • A proof is a sequence of statements forming an argument that shows that a theorem is true. The premises of the argument are axioms and previously proved theorems. ...
... • A theorem is a mathematical statement that can be shown to be true. • An axiom or postulate is an assumption accepted without proof. • A proof is a sequence of statements forming an argument that shows that a theorem is true. The premises of the argument are axioms and previously proved theorems. ...
Proofs • A theorem is a mathematical statement that can be shown to
... • A theorem is a mathematical statement that can be shown to be true. • An axiom or postulate is an assumption accepted without proof. • A proof is a sequence of statements forming an argument that shows that a theorem is true. The premises of the argument are axioms and previously proved theorems. ...
... • A theorem is a mathematical statement that can be shown to be true. • An axiom or postulate is an assumption accepted without proof. • A proof is a sequence of statements forming an argument that shows that a theorem is true. The premises of the argument are axioms and previously proved theorems. ...
