MAT 251 Discrete Mathematics

1. MAT 251 � Discrete Mathematics Logic and Proofs

2. Section 1.6 Introduction to Proofs Def: A conjecture is a statement that is being proposed to be a true statement, which is based on some examples or patterns or some intuition. Def: A theorem is a statement that can be shown to be true. To demonstrate that a theorem is true we use proofs.

3. Section 1.6 Introduction to Proofs Def: A proposition is a theorem with not such great significance. Def: An axiom/postulate is a statement we assume to be true without any proof.

4. Section 1.6 Introduction to Proofs Def: A less important theorem with partial results of a big theorem is called a lemma. Def: A proof is a valid argument that establishes the truth of a mathematical statement. A proof can use the hypotheses of the theorem, if any, axioms assumed to be true, and previously proven theorems such as lemmas and propositions and all terms must be well defined. Most of the proofs we will investigate are informal proofs. Def: A corollary is a theorem that is a direct consequence of a theorem that is established.

5. Section 1.6 Introduction to Proofs Most theorems are stated as implications or bi-implications. Example; If x > 0, then x2 > 0. |x| = 0 iff x = 0

6. Section 1.6 Introduction to Proofs Methods of Proofs: 1) Direct Proof 2) Proof by Contraposition 3) Proof by Contradiction (2 & 3 are indirect proofs) and more ? to come!

7. Section 1.6 Introduction to Proofs Direct Proof Given an implication p ? q a direct proof is constructed by assuming that p is true and showing that q must also be true. We use definitions, axioms and previously proven theorems in a valid sequence of arguments.

8. Section 1.6 Introduction to Proofs Def: The integer n is even if there exists an integer k such that n = 2k, and n is odd if there exists an integer k such that n = 2k + 1. (Note that integers are either even or odd, and no integer is both even and odd.)

9. Section 1.6 Introduction to Proofs Prove the Thm: �If n is an even integer, then n2 is even.� Pf: Let�s assume that n is an even integer. Then by definition, there exists an integer k such that n = 2k. By the property of equality, we can square both sides of this equation to get that n2 = (2k)2 = 4k2 = 2(2k2), which by definition is an even integer. Thus, we have proven that if n is an even integer, then n2 is even.

10. Section 1.6 Introduction to Proofs Prove the Thm: �If m and n are integers, m is odd and n is even then m+n is odd.�

11. Section 1.6 Introduction to Proofs Proof by Contraposition This proof uses the fact that the implication p ? q ? �q ? �p. In this proof, we take �q as our hypothesis and show using axioms, definitions and previously proven theorems with valid arguments that �p must follow.

12. Section 1.6 Introduction to Proofs Page 85/#16 Prove using Proof by Contraposition the Thm: �If m and n are integers and mn is even, then m is even or n is even.� (Note: p ? (q ? r) = � (q ? r) ? �p = (�q ? �r) ? �p

13. Section 1.6 Introduction to Proofs Page 85/#16 Prove using Proof by Contraposition the Thm: �If m and n are integers, mn is even, then m is even or n is even.� is equivalent to �If m and n are integers, m is odd and n is odd, then mn is odd.�

14. Section 1.6 Introduction to Proofs �If m and n are integers, m is odd and n is odd, then mn is odd.� Pf: Let�s assume that m and n are odd integers. Then by definition, there exists integer i and j, such that m = 2i+1 and n = 2j+1. Hence, mn = (2i+1)(2j+1) = 4ij + 2(i+j) + 1= 2(2ij + i +j) + 1, or mn is an odd integer.

15. Section 1.6 Introduction to Proofs Def: The real number r is rational if there exist integers p and q with q ? 0, such that r = p/q. A real number is not rational is called irrational.

16. Section 1.6 Introduction to Proofs Prove using Proof by Contraposition the Thm: �If x is irrational, then 1/x is irrational.�

17. Section 1.6 Introduction to Proofs Proof by Contradiction Suppose you have a proposition p that you want to prove is true. Also, suppose you can find a contradiction q such that �p ? q is a true statement. This can only happen if �p is false, or p is true.

18. Section 1.6 Introduction to Proofs Page 85/#18 Prove using the proof by contradiction the Thm: �If n is an integer and 3n+2 is even, then n is even.� (Note: �(p ? q) = p ? �q

19. Section 1.6 Introduction to Proofs �If n is an integer and 3n+2 is even, then n is even.� Pf: Let�s assume that 3n + 2 is even and n is an odd integer. Then we have that (3n+2) + n = 4n + 2 = 2(2n + 1), which is an even integer. However, that is a contradiction, since the sum of an even and odd integer is odd.

20. Section 1.6 Introduction to Proofs Prove that v3 is irrational.

21. Section 1.6 Introduction to Proofs The notes have been created with the use of Discrete mathematics and Its Applications, Sixth Edition by K. H. Rosen