Questions: Introduction to Proof by Contradiction

Question 1

In a proof by contradiction of 'There is no largest even number,' what do you assume?

• A. There is no largest even number
• B. There is a largest even number
• C. All even numbers are larger than all odd numbers
• D. The largest even number is 100

Answer: (B)

In proof by contradiction, you assume the negation of what you want to prove. The negation of 'there is no largest even number' is 'there is a largest even number.' You then show this leads to a contradiction — if N is the largest even number, then N + 2 is even and larger than N, contradicting the assumption.

Question 2

In a proof by contradiction, if you successfully derive a contradiction, the original statement you wanted to prove is false.

• T. True
• F. False

Answer: False

It is the opposite: deriving a contradiction means your ASSUMPTION (the negation of the original statement) is false. Since the negation is false, the original statement must be true. The contradiction proves the original statement, not disproves it.

Question 3

Prove by contradiction that the sum of a rational number and an irrational number is irrational.

Model answer: Assume for contradiction that a is rational, b is irrational, and a + b = c is rational. Since a and c are rational, c - a is rational (rationals are closed under subtraction). But c - a = b, so b is rational. This contradicts our assumption that b is irrational. Therefore a + b must be irrational.

The proof assumes the negation (the sum is rational) and uses closure properties of rational numbers to derive that b must be rational — directly contradicting the given that b is irrational. The contradiction forces us to reject the assumption, proving the original claim.