Properties of the Comparison Relations

There are six symbols used for comparison of numbers and other mathematical objects:

=		equal to		5 = 1 + 4 but not 5 = 4
≠		not equal to		5 ≠ 4 but not 5 ≠ 5
<		less than		4 < 5 but not 4 < 4
>		greater than		5 > 4 but not 5 > 5
≤		less than or equal to		4 ≤ 5, 4 ≤ 4, but not 5 ≤ 4
≥		greater than or equal to		5 ≥ 4, 5 ≥ 5, but not 4 ≥ 5

The six symbols describe possible relationships the numbers may stand in to each other. They have the following properties. I use the double arrow (⇒) as a shorthand for "imply", "if - then".

The relation "not equal" is not reflexive: for no number a, a ≠ a. It is also not transitive, for example, 3 ≠ 5 and 5 ≠ 1 + 2 but, nonetheless, 3 = 1 + 2.

Being less means, in particular, not being equal to, so that this relation is not reflexive: it is not true that, say, 5 < 5. Similarly,

And also

By the definition, a ≤ b means that either a < b or a = b. We summarize other links between different relations:

a ≤ b ⇒ a < b  or  a = b
a ≥ b ⇒ a > b  or  a = b
a < b ⇒ a ≤ b  and  a ≠ b
a > b ⇒ a ≥ b  and  a ≠ b

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