Answer:
a)
Given Statement - If x and y are a pair of consecutive integers, then x and y have opposite parity.
Proof by Contrapositive:
Assumed statement: Suppose that integers x and y do not have opposite parity.
Proven Statement: x and y are not a pair of consecutive integers.
Proof -
x = 2u₁ , y = 2u₂
Then
(x, x+1) = (2u₁ , 2u₁ + 1) = (Even, odd)
If y = 2u₁ + 1
Not possible
⇒x and y are not a pair of consecutive integers.
Hence proved.
Proof by Contradiction:
Assumed statement: Suppose x and y are not a pair of consecutive integers.
Proven Statement: Suppose x and y do not have opposite parity.
Proof -
If x and y are not a pair of consecutive integers.
⇒ either x and y are odd or even
If x and y are odd
⇒x and y have same parity
Contradiction
If x and y are even
⇒x and y have same parity
Contradiction
(b)
Proof by Contrapositive:
Assumed statement: Let n be an integer such that n is not odd (i.e. n is an even integer)
Proven Statement: n² is not odd (i.e n² is even)
Proof -
Let n is even
⇒n = 2m
⇒n² = (2m)² = 4m²
⇒n² is even
Hence proved.
Proof by Contradiction:
Assumed statement: Let n be an integer such that n² be odd.
Proven Statement: suppose that n is not odd (i.e n is even)
Proof -
Let n² is odd
⇒n² is even
⇒n² = 2m
⇒2 | n²
⇒2 | n
⇒n = 2x
⇒ n is even
Contradiction
