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

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.

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Thông Đạt 6 months 2021-07-17T07:32:33+00:00 1 Answers 6 views 0

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    2021-07-17T07:34:11+00:00

    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

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