CS2300-1.7
... • Direct Proof: Assume that p is true. Use rules of inference, axioms, and logical equivalences to show that q must also be true. Example: Give a direct proof of the theorem “If n is an odd integer, then n2 is odd.” Solution: Assume that n is odd. Then n = 2k + 1 for an integer k. Squaring both side ...
... • Direct Proof: Assume that p is true. Use rules of inference, axioms, and logical equivalences to show that q must also be true. Example: Give a direct proof of the theorem “If n is an odd integer, then n2 is odd.” Solution: Assume that n is odd. Then n = 2k + 1 for an integer k. Squaring both side ...
2 - DePaul University
... settings to draw conclusions about particular elements nor did they know what processes must be followed to establish the truth of universally (or even existentially) quantified statements. ...
... settings to draw conclusions about particular elements nor did they know what processes must be followed to establish the truth of universally (or even existentially) quantified statements. ...
![[Part 2]](http://s1.studyres.com/store/data/008795781_1-3298003100feabad99b109506bff89b8-300x300.png)
01-NumberTheoryslides
... associates, tools and enemies. • Applied adjectives associated with people such as friendly, perfect, natural, rational. • Disintegration of school almost occurred when they discovered that not all physical quantities were expressible as ratios of natural numbers. ...
... associates, tools and enemies. • Applied adjectives associated with people such as friendly, perfect, natural, rational. • Disintegration of school almost occurred when they discovered that not all physical quantities were expressible as ratios of natural numbers. ...
A Primer on Proving
... In a classroom setting, a problem that starts off with “prove this theorem” is an exercise in which students are assured that what they are working on can be proven by the word “theorem” in the problem statement. Neither teachers nor mathematicians call a statement “theorem” unless it’s been proved. ...
... In a classroom setting, a problem that starts off with “prove this theorem” is an exercise in which students are assured that what they are working on can be proven by the word “theorem” in the problem statement. Neither teachers nor mathematicians call a statement “theorem” unless it’s been proved. ...
Slide 1
... • Definition 2: The real number r is rational is there exist integers p and q with q0 such that r=p/q. A real number that is not rational is called irrational. • Ex.7: Prove that the sum of two rational numbers is rational. • Ex.8: Prove that if n is an integer and n2 is odd, then n is odd. ...
... • Definition 2: The real number r is rational is there exist integers p and q with q0 such that r=p/q. A real number that is not rational is called irrational. • Ex.7: Prove that the sum of two rational numbers is rational. • Ex.8: Prove that if n is an integer and n2 is odd, then n is odd. ...
Chapter Nine - Queen of the South
... are simply content just to appraise his conclusions which were revolutionary, to the point of being melancholic or disastrous in their revelation. Gödel purported to show that the axiomatic method in use from the times of Euclid and proven so efficacious through the Ages, possessed certain weaknesse ...
... are simply content just to appraise his conclusions which were revolutionary, to the point of being melancholic or disastrous in their revelation. Gödel purported to show that the axiomatic method in use from the times of Euclid and proven so efficacious through the Ages, possessed certain weaknesse ...
Predicate Calculus - SIUE Computer Science
... MATH 224 – Discrete Mathematics Proofs In the predicate calculus, it is not possible to use truth tables to prove most results since statements depend on one or more variables. This makes the job of proving results quite a bit more difficult. Would it be possible to use truth tables if the domain(s ...
... MATH 224 – Discrete Mathematics Proofs In the predicate calculus, it is not possible to use truth tables to prove most results since statements depend on one or more variables. This makes the job of proving results quite a bit more difficult. Would it be possible to use truth tables if the domain(s ...
File
... Deductive Reasoning ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ...
... Deductive Reasoning ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ...
Mathematical Reasoning (Part III)
... REMARK 13. If you use a proof by contradiction to prove that P ⇒ Q, your proof might begin with – Assume, to the contrary, that the statement P is true and the statement Q is false. or – By contradiction, assume, that the statement P is true and ¬Q is true. REMARK 14. If you use a proof by contradic ...
... REMARK 13. If you use a proof by contradiction to prove that P ⇒ Q, your proof might begin with – Assume, to the contrary, that the statement P is true and the statement Q is false. or – By contradiction, assume, that the statement P is true and ¬Q is true. REMARK 14. If you use a proof by contradic ...
Slides
... If the domain is finite, we can use the method of exhaustion, by simply trying every element Otherwise, we can use a direct proof ...
... If the domain is finite, we can use the method of exhaustion, by simply trying every element Otherwise, we can use a direct proof ...
1 Cardinality and the Pigeonhole Principle
... 4. This assertion is not provable in ZF, and is in fact equivalent to the axiom of choice (Hartog’s theorem). We omit the proof. Together we read these as saying that ≤ defines an order on the cardinal numbers. The first three points (which are true without the axiom of choice) show that ≤ gives a ...
... 4. This assertion is not provable in ZF, and is in fact equivalent to the axiom of choice (Hartog’s theorem). We omit the proof. Together we read these as saying that ≤ defines an order on the cardinal numbers. The first three points (which are true without the axiom of choice) show that ≤ gives a ...
Formal Methods Key to Homework Assignment 6, Part 3
... This is definitely not a counterexample. In the first place, C 6⊆ D. For example, {1} 6∈ D. In the second place, A∩C = ∅ and B∩D = {1, 2, 3, 4}. So A∩C ⊆ B∩D. 2. “Proof:” Let x ∈ A ∩ C. Then x ∈ A and x ∈ C. Since A ⊆ B and C ⊆ D, x ∈ B and x ∈ D. Therefore x ∈ B ∩ D, so A ∩ C ⊆ B ∩ D. This is fine. ...
... This is definitely not a counterexample. In the first place, C 6⊆ D. For example, {1} 6∈ D. In the second place, A∩C = ∅ and B∩D = {1, 2, 3, 4}. So A∩C ⊆ B∩D. 2. “Proof:” Let x ∈ A ∩ C. Then x ∈ A and x ∈ C. Since A ⊆ B and C ⊆ D, x ∈ B and x ∈ D. Therefore x ∈ B ∩ D, so A ∩ C ⊆ B ∩ D. This is fine. ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.