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Inequality symbol

From AoPSWiki

There are four symbols conventionally used to represent the notion of inequality.

If $a$ and $b$ are real numbers we write:

$a > b$ to mean that $a$ is strictly greater than $b$ (that is, $a$ cannot equal $b$ ).

$a \geq b$ to mean that $a$ is greater than or equal to (equivalently, "at least as large as") $b$ .

$a < b$ to mean that $a$ is strictly less than $b$

$a \leq b$ to mean that $a$ is less than or equal to $b$ .

We use a slash through an inequality symbol to represent that the given inequality does not hold. Thus for real numbers $a$ and $b$ ,

$a \not > b$ if and only if $a \leq b$
$a \not \geq b$ if and only if $a < b$
$a \not < b$ if and only if $a \geq b$
$a \not \leq b$ if and only if $a > b$
$\displaystyle a \neq b$ if and only if $a > b$ or $a < b$

These symbols are also frequently used to represent the order relation in a partially ordered set. Note that in this more general setting, it is not necessarily true that $a \not > b \Longleftrightarrow a \leq b$ , because it is also possible that $a$ and $b$ could be incomparable.

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Category: Notation

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