Download presentation

Presentation is loading. Please wait.

Published bySandra Stewart Modified over 4 years ago

1
Introduction to Proofs ch. 1.6, pg. 87,93 Muhammad Arief download dari http://arief.ismy.web.id http://arief.ismy.web.id

2
Proof Pembuktian tentang kebenaran suatu mathematical statement. Formal proof: utilize rule of inference all steps were supplied long and hard to follow suitable for computer Informal proof:steps maybe skip utilize assumption suitable for human http://arief.ismy.web.id

3
Proof Example: -Suppose you did commit the crime. -Then at the time of the crime, you would have had to be at the scene of the crime. -In fact, you were in a meeting with 10 people at that time, as they will testify. -This contradicts the assumption that you committed the crime. -Hence the assumption is false. http://arief.ismy.web.id

4
TOC Direct Proof Indirect Proof: Proof by Contraposition Proof by Contradiction Exhaustive Proof Proof Strategies http://arief.ismy.web.id

5
Proof Some Important Terminology: Theorem: a statement that can be shown to be true. Proposition: a less important theorem. Use “proof” to demonstrate that a theorem is true. Axiom: statements we assume to be true. http://arief.ismy.web.id

6
Theorem “For all positive real numbers x and y, if x > y, then x 2 > y 2 ” Mathematics convention: exclude the universal quantifier “If x > y, where x and y are positive real numbers, then x 2 > y 2 ” http://arief.ismy.web.id

7
Methods of Proving Theorems p q Proof: Demonstrate that q is true if p is true Methods: Direct Proofs Indirect Proofs: Proof by Contraposition Proof by Contradiction http://arief.ismy.web.id

8
Direct Proofs p q Direct Proof: -Assume that p is true -Use axiom, definition, rule of inference etc. -Show that q must also be true http://arief.ismy.web.id

9
Example Definition: 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. Give a direct proof of the theorem: “IF n is an odd integer, THEN n 2 is odd” Solution: -Assume that “n is odd” is true -Use axiom, definition, rule of inference etc -n = 2k + 1 (from definition) -n 2 = (2k + 1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1 -Conclusion: it is proved that n 2 is an odd integer http://arief.ismy.web.id

10
Example Definition: An integer a is a perfect square if there is an integer b such that a = b 2 Give a direct proof that: “IF m and n are both perfect squares, THEN nm is also a perfect square” Solution: -Assume that “m and n are both perfect square” is true -Use axiom, definition, rule of inference etc -There are integers s and t such that m = s 2 and n = t 2. -mn = s 2 t 2, mn = (st) 2 -Conclusion: it is proved that mn is a perfect square, because it is the square of st, which is an integer. http://arief.ismy.web.id

11
Proof by Contraposition Indirect Proof: proof that do not start with the hypothesis and end with the conclusion. Contraposition: p q is logically equivalent to ~q ~p. Proof by Contraposition: -Assume that ~q is true -Use axiom, definition, rule of inference etc. -Show that ~p must also be true http://arief.ismy.web.id

12
Example Prove that: “IF n is an integer and 3n + 2 is odd, THEN n is odd” Solution: -Assume that “n is even” is true -Use axiom, definition, rule of inference etc -n = 2k (from definition) -3n + 2 = 6k + 2 = 2 ( 3k + 1) -3n + 2 is even, and therefore not odd -Conclusion: it is proved by contraposition http://arief.ismy.web.id

13
Proof by Contradiction Contradiction: is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. q ~q ~(p q) p ~q Proof by Contradiction: -To prove that a statement p is true -Suppose the statement to be prove is false. Means its negation is true. -Show that this supposition leads logically to a contradiction, so the assumption is false. -Conclude that the statement to be proved is true. http://arief.ismy.web.id

14
Example Prove that: “IF 3n + 2 is odd, then n is odd” Proof by Contradiction: -p = 3n + 2 is odd, q = n is odd. -Assume that (p and ~q) is true. -Show that q is also true, OR -Show that ~p is also true. -~q = n is not odd = n is even -n = 2k -3n + 2 = 6k + 2 = 2 (3k + 1) = 2 t is even -~p is true -p and ~p are true contradiction http://arief.ismy.web.id

15
Example Prove that: “There is no greatest integer” Proof by Contradiction: -Assume that there is a greatest integer. -Then N n for every integer n. -Let M = N + 1, M is an integer -M > N contradiction http://arief.ismy.web.id

16
Example Prove that: “√2 is irrational” Proof by Contradiction: -p = √2 is irrational. -Assume that ~p is true. -~p = √2 is rational. -Definition: if q is rational, there exist integers a and b such that q = a / b, where a and b has no common factor. -Then there are integers a and b with no common factors so that √2 = a / b -2 = a 2 /b 2 -2b 2 = a 2 Definition of even: n = 2k -Then a 2 = is even a is also even -a = 2t 4t 2 = 2b 2 2t 2 = b 2 Definition of even: n = 2k -b 2 is even too b is also even -a and b has common factor of 2 contradiction -http://en.wikipedia.org/wiki/Irrational_number http://arief.ismy.web.id

17
Example Prove that: “1 + 3√2 is irrational” Proof by Contradiction: -p = 1 + 3√2 is irrational. -Assume that ~p is true. -~p = 1 + 3√2 is rational. -Definition: if q is rational, there exist integers a and b such that r = a / b, where a and b has no common factor. -1 + 3√2 = a/b -3√2 = a/b – 1 -√2 = (a – b) / (3b) -By definition: -A-b is integer -3 b is also integer -Then √2 is rational -Fact √2 is irrational http://arief.ismy.web.id

18
Exhaustive Proof Some theorems can be proved by examining a relatively small number of examples. Prove that (n+1) 3 3 n if n is a positive integer with n ≤ 4. Solution: n = 1; (1+1) 3 3 1 n = 2; (2+1) 3 3 2 n = 3; (3+1) 3 3 3 n = 4; (4+1) 3 3 4 http://arief.ismy.web.id

19
Proof Strategies Analyze what the hypothesis (premises) and conclusion mean. If it is a conditional statement: Try a Direct Proof Try a Proof by Contraposition Try a Proof by Contradiction If it has a relatively small number of domain: Try an Exhaustive Proof http://arief.ismy.web.id

20
Example Definition: The real number r is rational if there exists integers p and q with q ≠ 0 such that r = p/q. A real number that is not rational is called irrational. Prove that: “the sum of two rational number is rational.” Direct Proof: -Assume that “r and s are both rational numbers” is true -Use axiom, definition, rule of inference etc -There are integers p and q, with q ≠ 0, such that r = p/q. -There are integers t and u, with u ≠ 0, such that s = t/u. -r + s = p/q + t/u = (pu + qt) / qu -Conclusion: it is proved that r + s is rational. http://arief.ismy.web.id

21
Example Prove that: “IF n is an integer and n 2 is odd, THEN n is odd” Direct Proof: -Assume that “n 2 is odd” is true -Use axiom, definition, rule of inference etc -n 2 = 2k + 1 (from definition) -n = ±√ (2k + 1) -Conclusion: we can’t prove anything, try proof by contraposition http://arief.ismy.web.id

22
Example Proof by Contraposition: -Assume that “n is not odd” is true -Use axiom, definition, rule of inference etc -n is even -n = 2k (from definition) -n 2 = 4 k 2 = 2 (2 k 2 ), therefore n 2 is even (not odd) -Conclusion: it is proved by contraposition that “IF n is an integer and n 2 is odd, THEN n is odd” http://arief.ismy.web.id

23
Summary Direct Proof Proof by Contraposition Proof by Contradiction Exhaustive Proof Proof Strategies http://arief.ismy.web.id

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google