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Prove $\\var{2k}$ is even.

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By definition $\\var{2k}$ is even when there exists some integer $k$ such that $\\var{2k} = 2k$. It is easy to demonstrate that there is some integer $k$ with this property, because we can show exactly what it is: $k = $ [[0]].

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Prove $\\var{2*k1 + 1}$ is odd.

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By definition $\\var{2*k1 + 1}$ is odd when there exists some integer $k$ such that $\\var{2k1+1} = 2k+1$. It is easy to demonstrate that there is some integer $k$ with this property, because we can show exactly what it is: $k = $ [[0]].

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There exists some numbers which are even, and some numbers which are odd. But all numbers are even or odd. Formally we write this as

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$ \\forall n \\in \\mathbb Z$, $n$ is even or $n$ is odd.

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This is a statement about every possible integer. So we can not consider each integer individually and instead we consider an unspecified $n \\in \\mathbb Z$. There are an infinite number of individual existance proofs to do here, but we can reduce the workload to two possibilities by considering $n \\pmod 2$.

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If $n = $ [[0]] $\\pmod 2$ then $\\exists k \\in \\mathbb Z$ such that $n=2k$, hence $n$ is even.

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If $n = $ [[1]] $\\pmod 2$ then $\\exists k \\in \\mathbb Z$ such that $n = 1 + 2k$, hence $n$ is odd.

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Since these are the only possible values for $n \\pmod 2$, every integer $n$ is either even or odd.

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In this question we introduce the notion of existence (denoted $\\exists$) through the familiar concept of odd and even numbers. If $n\\in \\mathbb Z$ then we say

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$n$ is even when there exists some integer $k$ such that $n=2k$,

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$n$ is odd when there exists some integer $k$ such that $n=2k+1$.

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Prove the following statements.

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The purpose of these questions is to use the definition of even and odd.

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These are examples of constructive proofs, where we prove existence by constructing number with the required property.

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Not all proofs are constructive. For example, to prove that every integer is even or odd we rely upon the properties of arithmetic in $\\pmod 2$ to show that there must exist some number $k$ such that $n=2k$ or $n=2k+1$. But, in contrast with the above examples, we do not actually say what that value is.

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Intorduction to using the defnition to prove simple stamtements.

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