Equivalence of Definitions of Even Integer
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Theorem
The following definitions of the concept of Even Integer are equivalent:
Definition 1
An integer $n \in \Z$ is even if and only if it is divisible by $2$.
Definition 2
An integer $n \in \Z$ is even if and only if it is of the form:
- $n = 2 r$
where $r \in \Z$ is an integer.
Definition 3
An integer $n \in \Z$ is even if and only if:
- $x \equiv 0 \pmod 2$
where the notation denotes congruence modulo $2$.
Proof
$(1)$ if and only if $(2)$
By definition of divisor, $n$ is divisible by $2$ if and only if:
- $n = 2 r$
where $r \in \Z$.
Thus definition 1 is logically equivalent to definition 2.
$\Box$
$(2)$ if and only if $(3)$
By definition of congruence modulo $2$:
- $x \equiv y \pmod 2 \iff \exists r \in \Z: x - y = 2 r$
Setting $y = 0$:
- $x \equiv 0 \pmod 2 \iff \exists r \in \Z: x = 2 r$
Thus definition 2 is logically equivalent to definition 3.
$\blacksquare$