Reading a definition

A paper says: a whole number n is called even if there is a whole number k with n equal to twice k. Read slowly. The word being defined is even, and everything after the word if is the test.

A paper says: a whole number $n$ is called even if there is a whole number $k$ with $n=2k$. Read slowly: the name is even; the test follows the word if.

$E:=\{n\in \mathbb{Z}\mid \exists k\in \mathbb{Z}:n=2k\}$

Check the test with numbers. Eight is twice four, so eight is even. Seven is not twice any whole number, so seven is not even. The test sorts numbers into two piles.

Check the test. $8=2\cdot 4$, so $8$ is even. $7\ne 2k$ for every whole $k$, so $7$ is not.

$8=2\cdot 4\Rightarrow 8\in E$. $\nexists k:7=2k\Rightarrow 7\notin E$.

Look at what the test does not say. It does not say that k must be positive, so zero, which is twice zero, is even. It says nothing about fractions: three halves is not a whole number, so the test does not apply to it.

What the test omits: no sign for $k$, so $0=2\cdot 0$ is even. Nothing on fractions: $\frac{3}{2}\notin \mathbb{Z}$, so it is not tested.

$0=2\cdot 0\Rightarrow 0\in E$. $\frac{3}{2}\notin \mathbb{Z}$: untested.

Now use the definition. If n is even, then n plus two is even: n is twice k, so n plus two is twice the number k plus one.

Use it: $n$ even $\Rightarrow n+2=2k+2=2(k+1)$, even.

$n\in E\Rightarrow n+2=2(k+1)\in E$.

The lesson: a definition is a test. Find the name, find the test, try numbers on both sides of the line, and only then use it.

The lesson: a definition is a test. Name, test, numbers on both sides, then use.

Name $E$. Test $\exists k$. Try $8\in E$, $7\notin E$. Use: $n+2\in E$.