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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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