Problem 10 of Review Questions of Chapter 1

a) A Direct Proof considers a conditional statement p . q and assumes that p is true and uses known axioms, and previously proven theorems to show that q must also be true.

A Proof by Contraposition uses the rule of Contrapostion ( p . q = -q . -p) to prove p . q You take the conclusion and force it to be false or .q. Then you show that follows the .q will lead to .p by using axioms and theorems.

A Proof by Contradiction seeks to find a q such that .p . q is true. When q is false it follows that .p is false so p is true.

b) If n is even then n + 4 is even Direct Proof n is even if n = 2k - From the definition of an even number n + 4 = 2k + 4 = 2(k+2) Thus n + 4 is an even number because 2(k+2) is 2 times and integer. Proof by Contraposition An odd number is an even number plus one so if n + 4 = q then n + 5 = -q n + 5 = 2k + 1 n = 2k - 4 n is always even for all values of k so -q came to a conclusion -p or -q --> -p Thus p --> q is true Proof by Contradiction -p is n = 2k + 1 Now we sub it into q and see if it is odd or even n + 4 = 2k + 1 + 4 = 2k +5 q is always odd and thus false thus .p is false and proves the proposition p.