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mixer [17]
3 years ago
12

If k is an integer, what is k + 1 /2 ! ? Why?

Mathematics
1 answer:
Amanda [17]3 years ago
8 0

If k\in\mathbb{Z} than k+1\in\mathbb{Z} however any \mathbb{Z} divided by 2 is in \mathbb{Q}.

But it is not so simple.

If k is odd then k+1 is even but when even number is divided by 2, you get an odd number which is in \mathbb{Z}.

However if k is even then k+1 is odd and when we divide that by 2, you get a number like for example \frac{5}{2} which is not in \mathbb{Z} but rather inside \mathbb{Q}. There are fortunately no irrational numbers.

There is also a problem with zero, zero is neither odd nor even but still an integer. If your k happens to be -1, k+1=0, and zero divided by 2 is still just 0 which is in \mathbb{Z}.

So to sum up,

If k is odd and not -1, then \frac{k+1}{2}\in\mathbb{Z}.

Or to put it in math \forall k\in\mathrm{odd}-\{-1\}\cup\{0\}\implies\frac{k+1}{2}\in\mathbb{Z}.

If k is even or -1, then \frac{k+1}{2}\in\mathbb{Q}.

Or to put it in math \forall k\in\mathrm{even}\cup \{-1\}\implies\frac{k+1}{2}\in\mathbb{Q}.

<u>Notes</u>:

\mathbb{Z} is a set of integers.

\mathbb{Q} is a set of rational numbers.

\mathbb{Z}\subset\mathbb{Q} means integers are a subset of rational numbers, that is, every integer is a also a fraction, however not every fraction is an integer.

Hope this helps :)

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Applying the inclusion-exclusion formula

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