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Let n e Z. Use a proof by contraposition to show that if lin - 5 is an odd integer, then n is an even integer.

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Sure, we can break this down into several steps.

The contrapositive of the assertion "if lin - 5 is an odd integer, then n is an even integer" is "if n is not an even integer, then lin - 5 is not an odd integer."

Contrapositive reasoning enables us to handle this problem, so we will start with the assumption that n is not an even integer. The set of all integers is composed of two disjoint sets: odd and even integers. Therefore, if n is not even, n must be an odd integer.

An odd integer can be expressed in the form 2k + 1, where k is any integer. So, we can rewrite n as 2k + 1.

Now, if we substitute n = 2k + 1 into lin - 5, we obtain lin - 5 = l(2k + 1) - 5.

Simplifying this expression gives us l(2k + 1) - 5 = 2lk + l - 5.

However, for a number to be odd, it should be of the form 2p + 1, where p is any integer. As we can see, 2lk + l - 5 does not fit this form, and so it is not an odd number.

Therefore, if n is not an even integer, then lin - 5 cannot be an odd integer.

So, by contraposition, we have shown that if lin - 5 is an odd integer, then n must indeed be an even integer.

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