1. 1 Introduction to Abstract Mathematics Proof Methods , , ~, ,  Instructor: Hayk Melikya melikyan@nccu.edu Purpose of Section:Most theorems in mathematics take the form of an implication P  Q or as a biconditional P  Q, where the biconditional can be verified by proving both P  Q and Q  P. We will study a variety of ways of proving P  Q including a direct proof, proof by contrapositive, and three variations of proof by contradiction, including proof by reductio ad absurdum

2. 2 Introduction to Abstract Mathematics Many theorems in mathematics have the form of an implication P  Q, where one assumes the validity of P, then with the aid of existing mathematical facts and theorems as well as laws of logic and reasoning, arrive at the conclusion Q. Although the goal is always to “go from P to Q,” is more than one valid way of achieving this goal. Five equivalent ways to proving the implication P  Q are shown in next

3. 3 Introduction to Abstract Mathematics Five forms

4. 4 Introduction to Abstract Mathematics Example 1 (Direct Proof) If n is an odd natural number, then n 2 is odd. Proof: An integer n is called odd if it is of the form n = 2k +1 for some integer k, then since we assumed n odd, we can write n = 2k +1. Squaring gives n 2 = (2k +1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k ) + 1 Since k is a natural number we know s = 2k 2 + 2k is a natural number and so n 2 = 2s +1 which proves that n 2 is odd.

5. 5 Introduction to Abstract Mathematics You should remember the symbols used to denote for each set such as increasing collection of set N, Z, Q, R, C since we will be referring to these sets in the remainder of the book.

6. 6 Introduction to Abstract Mathematics Direct Proof [P  Q] A direct proof starts with the given assumption P and uses existing facts to establish the truth of the conclusion Q. More formally: Let H 1, H 2, … H k, P and Q be a propositional expressions then H 1, H 2, … H k ├ P  Q if and only if H 1, H 2, … H k, P ├ Q. The Rule of Direct Proof [DP] ( often called Deduction Theorem) can be shown to be true by use of Exportation (see the theorem ): H 1, H 2, … H k,  (P  Q)  (H 1  H 2  …  H k  P )  Q.

7. 7 Introduction to Abstract Mathematics Example: Prove (P  Q)  (R  S)├ P  R. Proof: 1. (P  Q)  (Q  R) Hyp (Goal P  R) 2. P Assum 3. P  Q 2 Add 4. Q  R 2, 1 MP 5. R 4 S 6. P  R. 25 DP END

8. 8 Introduction to Abstract Mathematics We can use this approach to write paragraphs proofs also. Always follow the following steps while writing paragraph proofs: 1. Identify hypothesis and conclusion of implication, 2. assume the hypothesis 3. translate the hypothesis (replace more useable equivalent form), 4. write comment regarding what must be shown, 5. translate the comment (replace more useable equivalent form) 6. deduce the conclusion

9. 9 Introduction to Abstract Mathematics Example: Let x be an integer. Prove that if x is an odd integer then x + 1 is even integer. Let P:=” x is odd integer” (hypothesis) and Q := “ x + 1 is even integer ” (conclusion). What we want ├ P  Q Assume P: Translate: by definition x is even if x = 2k +1 for some intege r k Then x + 1 = (2k + 1) +1 = 2k + 2 = 2(k + 1) Indirect proofs2 refer to proof by contrapositive or proof by contradiction which we introduce here

10. 10 Introduction to Abstract Mathematics Indirect Proofs: Indirect proofs refer to proof by contrapositive or proof by contradiction which we introduce next. A contrapositive proof or proof by contrapositive for conditional proposition P  Q one makes use of the tautology (P  Q)  (  Q   P). Since P  Q and  Q   P are logically equivalent we first give a direct proof of  Q   P and then conclude that P  Q. One type of theorem where proof by contrapositive is often used is where the conclusion Q states something does not exist.

11. 11 Introduction to Abstract Mathematics Here is the structure of contrapositive proof of P  Q Proof: Assume,  Q,. Therefore,  P. Thus,  Q   P Therefore, P  Q. END

12. 12 Introduction to Abstract Mathematics Contradictions -1: Let H 1, H 2, … H k and Q be a propositional expressions then H 1, H 2, … H k ├ Q if and only if H 1, H 2, … H k  Q ├ P   P for some propositional expression P. This is also known as proof by contradiction or reductio ad absurdum.

13. 13 Introduction to Abstract Mathematics Contradictions-2 [ ( P   Q)   P or (P   Q)  Q ] A proof by contradiction is based on the tautology P  (  P  (Q  Q)). Here the hypotheses P to be true but the conclusion Q false, and from this reach some type of contradiction, either contradicting the assumption P or contradicting the denial  Q

14. 14 Introduction to Abstract Mathematics Reductio ad absurdum [ ( P   Q)  R   R ]. Reductio ad absurdum is another form of proof by contradiction. After assuming P is true and Q is not true, one seeks to prove an absurd result (like 1 = 0 or x 2 < 0 ), which we denote by R   R.

15. 15 Introduction to Abstract Mathematics Not Mathematical Proofs: ▶ The proof is so easy we’ll skip it. ▶ Don’t be stupid, of course it’s true! ▶ It’s true because I said it’s true! ▶ Oh God let it be true! ▶ I have this gut feeling. ▶ I did it last night. ▶ I define it to be true! You can think of a few yourself.

16. 16 Introduction to Abstract Mathematics Examples: Example 6 ( Proof by Contrapositive ) : Let n be a natural number. If n 2 is odd so is n. Example 7 (Proof by Contradiction): is irrational number.

17. 17 Introduction to Abstract Mathematics Statement forms in Mathematics

18. 18 Introduction to Abstract Mathematics

19. 19 Introduction to Abstract Mathematics