Discrete Structures Introduction to Proofs

1. Discrete StructuresIntroduction to Proofs Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore. mhumayoun@ciitlahore.edu.pk https://sites.google.com/a/ciitlahore.edu.pk/dstruct/ Some material is taken from Dr. Atif’s slides

2. Terminology • Theorem: a statement that can be shown true. Sometimes called facts. • Proof: Demonstration that a theorem is true. • Axiom: A statement that is assumed to be true. • Lemma: a less important theorem that is useful to prove a theorem. • Corollary: a theorem that can be proven directly from a theorem that has been proved. • Conjecture: a statement that is being proposed to be a true statement.

3. Stating Theorems • Theorem. If , where x and y are positive real numbers, then . • Theorem. For all positive real numbers x and y, if , then .

4. Methods of Proving Theorems

5. Trivial Proofs • Consider an implication: • If it can be shown that p is true, then the implication is always true • By definition of an implication • Note that you are showing that the conclusion is true

6. Trivial Proof Example Consider the statement: • If you are in CSC102 then you are a student. • Since all people in CSC102 are students, the implication is true.

7. Vacuous proofs • Consider an implication: • If it can be shown that is false, then the implication is always true. • By definition of an implication • Note that you are showing that the hypothesis is false

8. Example Consider the statement: • Every student snooze during class when he is tired. • Rephrased: If a student is tired then he snoozes during class • Since there is no such student who is snoozing, (the hypothesis is false), the implication is true

9. Methods of Proving Theorems Important ones

10. Methods of Proving Theorems Direct Proofs • Consider an implication: • If p is false, then the implication is always true • Thus, show that if p is true, then q is true • To perform a direct proof, assume that p is true, and show that q must therefore be true

11. Example Direct Proof Theorem. Show that the square of an even number is an even number. For every number , if is even, then is even.

12. Example Direct Proof Theorem. Show that the square of an even number is an even number. For every number , if is even, then is even.

13. Example Direct Proof Theorem. Show that the square of an even number is an even number. For every number , if is even, then is even. Proof. Usual convention: Universal instantiation is not explicitly used and it is directly showed that implies .

14. Example Direct Proof Theorem. Show that the square of an even number is an even number. For every number , if is even, then is even. Proof. Usual convention: Universal instantiation is not explicitly used and it is directly showed that implies . Assume is even. Thus, , for some (definition of even numbers). As is 2 times an integer, is thus even.

15. Example Direct Proof Theorem. Show that the square of an even number is an even number. For every number , if is even, then is even. Proof. Usual convention: Universal instantiation is not explicitly used and it is directly showed that implies . Assume is even. Thus, , for some (definition of even numbers). As is 2 times an integer, is thus even.

16. Proof by Contraposition(or Indirect proofs) • If the antecedent is false, then the contrapositive is always true • Thus, show that if is true, then is true • To perform an indirect proof, do a direct proof on the contrapositive

17. Indirect proof example Theorem. If is an odd integer then is an odd integer Proof (by contrapositive). We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold. Contrapositive: If is an even integer, then is an even integer.

18. Indirect proof example Theorem. If is an odd integer then is an odd integer Proof (by contrapositive). We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold. Contrapositive: If is an even integer, then is an even integer. Assume that is even. Then by the definition of even numbers: for some integer .

19. Indirect proof example Theorem. If is an odd integer then is an odd integer Proof (by contrapositive). We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold. Contrapositive: If is an even integer, then is an even integer. Assume that is even. Then by the definition of even numbers: for some integer . Thus Since is 2 times an integer, it is even. Hence our proof by contraposition succeeds.

20. Selecting a Proof method Theorem. If is an integer and is odd, then is even. Proof. (Via direct proof). Assume that is odd. Then by the definition of odd numbers for some integer k. Thus … ??? Direct proof doesn’t seem to work.

21. Selecting a Proof method Theorem. If is an integer and is odd, then is even. Proof. (Via direct proof). Assume that is odd. Then by the definition of odd numbers for some integer k. Thus … ??? Direct proof doesn’t seem to work.

22. Selecting a Proof method Theorem. If is an integer and is odd, then is even. Indirect proof. Contrapositive: If is odd, then is even. Assume is odd, and show that is even. By the definition of odd numbers for some integer . As is times an integer, it is even

23. Selecting a Proof method Theorem. If is an integer and is odd, then is even. Indirect proof. Contrapositive: If is odd, then is even. Assume is odd, and show that is even. By the definition of odd numbers for some integer . As is times an integer, it is even

24. Selecting a Proof method Theorem. If is an integer and is odd, then is even. Indirect proof. Contrapositive: If is odd, then is even. Assume is odd, and show that is even. By the definition of odd numbers for some integer . As is times an integer, it is even

25. Selecting a Proof method Theorem. If is an integer and is odd, then is even. Indirect proof. Contrapositive: If is odd, then is even. Assume is odd, and show that is even. By the definition of odd numbers for some integer . As is times an integer, it is even

26. Proof by contradiction (another type of indirect proofs) • Given a statement of the form • Assume p is true and q is false • Assume • Then prove that cannot occur • A contradiction exists

27. Proof by contradiction example Theorem. If is an integer and is odd, then is even. Rephrased: If is odd, then is even Proof. Assume p is true and q is false (Assume . Assume ) Assume that is odd, and is odd.

28. Proof by contradiction example Theorem. If is an integer and is odd, then is even. Rephrased: If is odd, then is even Proof. Assume p is true and q is false(Assume . Assume ) Assume that is odd, and is odd. By the definition of odd numbers for some integer k.

29. Proof by contradiction example Theorem. If is an integer and is odd, then is even. Rephrased: If is odd, then is even Proof. Assume p is true and q is false (Assume . Assume ) Assume that is odd, and is odd. By the definition of odd numbers for some integer k.

30. Proof by contradiction example Theorem. If is an integer and is odd, then is even. Rephrased: If is odd, then is even Proof. Assume p is true and q is false (Assume . Assume ) Assume that is odd, and is odd. By the definition of odd numbers for some integer k. As is 2 times an integer, it must be even. Contradiction!

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