Question

For each statement, write what would be assumed and what would be proven in a proof by contrapositive of the statement. Then write what would be assumed and what would be proven in a proof by contradiction of the statement.

a. If x and y are a pair of consecutive integers, then x and y have opposite parity.
b. For all integers n, if n² is odd, then n is also odd.

Answer 1

a.)

Statement - If x and y are a pair of consecutive integers, then x and y have opposite parity.

Contra-positive statement - If x and y have same parity, then x and y can not be a pair of consecutive integers.

Assumed - x and y have same parity

To prove - x and y can not be a pair of consecutive integers.

Contradiction statement - If x and y are a pair of consecutive integers and they have same parity then then it is not possible.

Assumed - x and y are a pair of consecutive integers and they have same parity

To prove - It is not possible

Proof - Let x and y are even numbers

⇒x = 2a , y = 2b

then , y - x = 2b - 2a = 2(b - a) is alteast 0 if x ≠y

Contradiction.

b.)

Statement - For all integers n, if n² is odd, then n is also odd.

Contrapositive statement - If n is even , then n² is even

assumed - n is even

To prove - n² is even

Contradiction - If n² is odd and n is even then it is not possible

Assumed - n² is odd and n is even

To prove - Our assumption is not true

Proof - As n is even

⇒n = 2a

⇒n² = (2a)² = 4a²

⇒n² is even

Contradiction.