Proof by Contradiction CS 270 Math Foundations of CS Jeremy Johnson

Outline

Decimals and Fractions

Repeating Decimals

Theorem. A number r is rational iff it has a terminating or repeating decimal expansion Proof  If r = a/b perform long division to compute the decimal expansion. At each step divide what is left by b, m = qb + r, 0 ≤ r < b. There are b possible remainders. If r = 0 the expansion is terminating. If r has occurred previously the expansion is repeating. After at most b steps one of these must happen.

Repeating Decimals

Decimal Expansion of sqrt(2)

x0 := 1.0; x1 := 2.0; n := 20; for i from 1 to n do x := (x0+x1)/2; if x^2 > 2 then x1 := x; else x0 := x; end if; end do;

Decimal Expansion of sqrt(2)  Does the expansion terminate or repeat?  Maybe it doesn’t?  How long should I look?  Maybe it’s not rational?

Proof that sqrt(2) is not Rational

Negation Rules

Introduction and elimination rules Double negation     e 

Proof by Contradiction Negation elimination called proof by contradiction Assume  and derive a a contradiction

Exercise Prove that  A  A and A   A

Law of the Excluded Middle 1  (p  p) assumption 2Assumption 3 (p  p) 4 5  p 6 p  p 7 8

De Morgan’s Law  (P  Q)   P   Q 1 (P  Q)(P  Q) premise 2assumption 3 P  QP  Q  i PP 6Q 7 P  QP  Q  i P  QP  Q  i 5,9

De Morgan’s Law  (P  Q)   P   Q 1 P  QP  Q premise 2  e1 1 3  e2 1 4assumption 5P 6  e 2,5 7Q  i2 6 8  e 3,7 9  e 4,5-6, (P  Q)(P  Q)  i 4-9

Exercise Prove  (P  Q)   P   Q

Exercise Prove  P   Q   (P  Q)