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Discrete Structures

Discrete Structures. Chapter 4: Elementary Number Theory and Methods of Proof 4.6 Indirect Argument: Contradiction & Contraposition.

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Discrete Structures

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  1. Discrete Structures Chapter 4: Elementary Number Theory and Methods of Proof 4.6 Indirect Argument: Contradiction & Contraposition Reductio ad absurdum is one of a mathematician’s finest weapons. It is a far finer gambit: a chess player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game. – G. H. Hardy, 1877 – 1947 4.6 Indirect Argument: Contradiction & Contraposition

  2. Method of Proof by Contradiction • Suppose the statement to be proved is false. That is, suppose that the negation of the statement is true. • Show that this supposition leads logically to a contradiction. • Conclude that the statement to be proved is true. 4.6 Indirect Argument: Contradiction & Contraposition

  3. Theorems • Theorem 4.6.1 There is no greatest integer. • Theorem 4.6.2 There is no integer that is both even and odd. • Theorem 4.6.3 The sum of any rational number and any irrational number is irrational. 4.6 Indirect Argument: Contradiction & Contraposition

  4. Example – pg. 206 #’s 10 & 14 • Prove the statement by contradiction. • The square root of any irrational number is irrational. 14. For all prime numbers a, b, and c, a2 + b2 c2. 4.6 Indirect Argument: Contradiction & Contraposition

  5. Example • Prove by contradiction that the is irrational. 4.6 Indirect Argument: Contradiction & Contraposition

  6. Method of Proof by Contraposition • Express the statement to be proved in the form x in D, if P(x) then Q(x). • Rewrite this statement in the contrapositive form x in D, if Q(x) is false then P(x) is false. • Prove the contrapositive by direct proof. • Suppose x is an element of Ds.t. Q(x) is false. • Show that P(x) is false. 4.6 Indirect Argument: Contradiction & Contraposition

  7. Example – pg. 206 # 19 • Prove the statement by contraposition. • If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10. 4.6 Indirect Argument: Contradiction & Contraposition

  8. Example – pg. 206 #’s 25 & 26 • Prove the statements in two ways: a) by contraposition and b) contradiction. • For all integers n, if n2 is odd then n is odd. • For all integers a, b, and c, if a bc thena b. (Recall that  means does not divide.) 4.6 Indirect Argument: Contradiction & Contraposition

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