Indirect Argument Contradiction and Contraposition Lecture 17 Section

Form of Proof by Contraposition Theorem: p q. ¢ This is logically equivalent to

Benefit of Proof by Contraposition If p and q are “negative” statements, then p

Example: Proof by Contraposition ¢ ¢ Theorem: An irrational number plus 1 is irrational.

Form of Proof by Contradiction Theorem: p q. ¢ Outline of the proof of

Benefit of Proof by Contradiction ¢ The statement r may be any statement whatsoever

Proof by Contradiction ¢ Theorem: For all integers n 0, 2 + n is

Example Theorem: If x is rational and y is irrational, then x + y

Contraposition and Compound Hypotheses Often a theorem has the form (p q) r. ¢

Contraposition and Compound Hypotheses ¢ More generally, (p 1 p 2 … pn) q

Contradiction vs. Contraposition Sometimes a proof by contradiction “becomes” a proof by contraposition. ¢

Contradiction vs. Contraposition Would this be a proof by contradiction or proof by contraposition?

Contradiction vs. Contraposition? ¢ ¢ Theorem: If x is irrational, then –x is irrational.

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Indirect Argument: Contradiction and Contraposition Lecture 17 Section 3. 6 Mon, Feb 19, 2007

Form of Proof by Contraposition Theorem: p q. ¢ This is logically equivalent to q p. ¢ Outline of the proof of theorem: ¢ Assume q. l Prove p. l Conclude that p q. l ¢ This is a direct proof of the contrapositive.

Benefit of Proof by Contraposition If p and q are “negative” statements, then p and q are “positive” statements. ¢ We may be able to give a direct proof that q p more easily that we could give a direct proof that p q. ¢

Example: Proof by Contraposition ¢ ¢ Theorem: An irrational number plus 1 is irrational. Restate as an implication: Let r be a number. If r is irrational, then r + 1 is irrational. Restate again using the contrapositive: Let r be a number. If r + 1 is rational, then r is rational. Restate in simpler form: Let r be a number. If r is rational, then r – 1 is rational.

Form of Proof by Contradiction Theorem: p q. ¢ Outline of the proof of theorem : ¢ Assume (p q). l This is equivalent to assuming p q. l Derive a contradiction, i. e. , conclude r r for some statement r. l Conclude that p q. l

Benefit of Proof by Contradiction ¢ The statement r may be any statement whatsoever because any contradiction r r will suffice.

Proof by Contradiction ¢ Theorem: For all integers n 0, 2 + n is irrational.

Example Theorem: If x is rational and y is irrational, then x + y is irrational. ¢ That is, if ¢ p: x is rational l q: y is irrational l r: x + y is irrational, l ¢ Then l (p q) r.

Contraposition and Compound Hypotheses Often a theorem has the form (p q) r. ¢ This is logically equivalent to r (p q). ¢ However, it is also equivalent to p (q r). so it is also equivalent to p ( r q). ¢

Contraposition and Compound Hypotheses ¢ More generally, (p 1 p 2 … pn) q is equivalent to (p 1 p 2 … pn – 1) ( q pn).

Example Theorem: If x is rational and y is irrational, then x + y is irrational. ¢ Proof: ¢ Suppose x is rational. l Suppose also that x + y is rational. l Then y = (x + y) – x is the difference between rationals, which is rational. l Thus, if y is irrational, then x + y is irrational. l

Contradiction vs. Contraposition Sometimes a proof by contradiction “becomes” a proof by contraposition. ¢ Here is how it happens. ¢ To prove: p q. l Assume (p q), i. e. , p q. l Using q, prove p. l Cite the contradiction p p. l Conclude that p q. l

Contradiction vs. Contraposition Would this be a proof by contradiction or proof by contraposition? ¢ Proof by contraposition is preferred. ¢

Contradiction vs. Contraposition? ¢ ¢ Theorem: If x is irrational, then –x is irrational. Proof: l l l Suppose that x is irrational and that –x is rational. Let -x = a/b, where a and b are integers. Then x = -(a/b) = (-a)/b, which is rational. This is a contradiction. Therefore, if x is irrational, then -x is also irrational.