Elementary Number Theory and
Methods of Proof

Indirect Argument

• Argument by contradiction
• Illustration in proof of innocence
– Suppose I did commit the crime. Then at the time of the crime, I would have had to be at the scene of the crime.
– In fact, at the time of the crime, I was meeting with 20 people far from the crime scene, as they will testify.
– This contradicts the assumption that I committed the crime, since it is impossible to be in two places at one time. Hence, that assumption is false.

• Direct Proof
– start with hypothesis of a statement and make one deduction after another until the conclusion is reached.
• Indirect Proof (argument by contradiction)
– show that a given statement is not true leads to the contradiction.

• Use proof by contradiction to show that there is no greatest integer.
– Starting point: Suppose not. Suppose that there is a greatest integer, N. N≥n for all integers.
– To Show: This supposition leads logically to a contradiction.
– Proof:
• Suppose not. Suppose that there is a greatest integer N.
• N ≥ n for every integer n. Let M = N + 1. M is an integer under the addition property of integers.
• Thus M > N. N is not the greatest integer; therefore a contradiction.
• Theorem 3.6.1

• Theorem 3.6.2
– There is no integer that is both even and odd
– Proof:
• Suppose not. That is, suppose there is an integer n that is both even and odd.
• n = 2a (even) and n = 2b + 1 (odd)
• 2a = 2b + 1, 2a – 2b = 1,
• 2(a – b) = 1
• (a – b) = ½
• Since a and b are integers then a – b should result in an integer. Thus a-b is integer and a-b is not, contradiction.

• Theorem 3.6.3
– The sum of any rational number and any irrational number is irrational
– Proof:
• Suppose not. Suppose there is a rational number r and irrational number s such that r + s is rational.
• r = a/b (definition rational)
• r + s = c/d, for integers a,b,c,d with b≠0 and d≠0
• a/b + s = c/d, s = c/d – a/b
• s = (bc – ad)/bd (integer)
• hence, s is quotient of integers and therefore, rational
• If s is rational it contradicts supposition that it is irrational

• Argument by contrapositive is based on the equivalence of the statement and contrapositive.
• If the contrapositive is true then the statement is true.

• Method of Proof
1. Express the statement to be provided in the form: ∀ x in D, if P(x) then Q(x)
2. Rewrite the statement in the contrapositive form: ∀ x in D, if Q(x) is false then P(x) is false
3. Prove the contrapositive by a direct proof.
1. Suppose x is a (particular but arbitrarily chosen) element of D such that Q(x) is false.
2. Show that P(x) is false

• If the square of an integer is even, then the integer is even.
– Prove that for all integers n, if n 2 is even then n is even.
– Contrapositive: ∀ integers n, n is not even then n 2 is not even.
– Proof:
• Suppose n is any odd integer. n = 2k + 1
• n 2 = (2k + 1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1
• n 2 = 2r + 1 (by definition n 2 is odd)
• hence, the contrapositive is true therefore the statement must be true—”if n is even then n 2 is even”