Definition:Parity of Integer

From ProofWiki
Jump to navigation Jump to search

Definition

Let $z \in \Z$ be an integer.

The parity of $z$ is whether it is even or odd.


Even Parity

An integer $z$ is of even parity if and only if:

$z = 2 n$

for some $n \in \Z$.


Odd Parity

An integer $z$ is of odd parity if and only if:

$z = 2 n + 1$

for some $n \in \Z$.


Same Parity

Two integers $z_1$ and $z_2$ have the same parity if and only if either:

$z_1$ and $z_2$ are both even

or:

$z_1$ and $z_2$ are both odd.


Opposite Parity

Two integers $z_1$ and $z_2$ have opposite parity if and only if either:

$z_1$ is even and $z_2$ is odd

or:

$z_2$ is even and $z_1$ is odd.


Also defined as

Some sources define parity as a property of a pair of integers $\set {z_1, z_2}$ thus:

If $z_1$ and $z_2$ are either both even or both odd, $z_1$ and $z_2$ have even parity
If $z_1$ is even and $z_2$ is odd, then $z_1$ and $z_2$ have odd parity.


Also see

  • Results about parity of integers can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Parity_of_Integer&oldid=708174"