Is this discrete math question correct? Prove the following is true. If 7n+3 is odd, then n is even. *This is a proof by contrapositive?

My answer: Suppose n is odd.

So n= 2k+1 for some integer k.

7n+3=7(2k+1)+3= 14k+10 = 2(7k+5)

By definition, 7n+3 is even.

Since the contrapositive is true, that implies the original statement is also true.

Is this correct?

4 Answers

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  • 3 years ago
    Favourite answer

    Yes.

    The contrapositive of "if 7n + 3 is odd, then n is even" is "if n is odd, then 7n + 3 is even."

    You have proven the contrapositive to be true, therefore the original statement is true.

    {Another way to prove the statement without using contrapositives is:

    (An odd number) minus (an odd number) equals (an even number).

    If 7n + 3 is odd, then (7n + 3) - 3 = an even number.

    Thus, 7n + 3 - 3 = 7n is an even numer.

    (An odd number) times (an odd number) = (an odd number).

    Therefore, if 7n is even, then n cannot be odd and must be even.}

  • 3 years ago

    The only thing missing is that it must also state that n is an integer in the original statement. Otherwise the statement isn't true.

    The contrapositive of "If 7n+3 is odd, then n is even" would be "if n is not even then 7n+3 is not odd"

    But what if n = 1/7? n is not even but 7n+3 is odd.

  • Anonymous
    3 years ago

    Yep, your steps are all correct for a contrapositive proof.

  • 3 years ago

    (7n + 3) is odd → then n is even ?

    (7n + 3) ← if it's odd, you can divide it by 2

    = (7n + 3)/2

    = (7n/2) + (3/2)

    = (7n/2) + [(2 + 1)/2]

    = (7n/2) + (2/2) + (1/2)

    = (7n/2) + 1 + (1/2) → if n is even, you can divide it by 2

    = [7 * (n/2)] + 1 + (1/2) → if n is even, you can divide it by 2 → so (n/2) is an integer

    = [7 * integer] + 1 + (1/2) → you can say that [7 * integer] is too an integer

    = integer + 1 + (1/2) → and you can see that: integer + 1 is too an integer

    = integer + (1/2) → but you know that 1 is not divisible by 2

    As [integer + (1/2)] is not divisible by 2, so can say that: [integer + (1/2)] is odd

    → then: (7n + 3) is odd

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