# Properties of the Comparison Relations

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

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".

Equal to (=)

Not equal to (≠)

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.

Less than (<)

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,

Greater than (>)

Less than or equal to (≤)

And also

Greater than or equal to (≥)

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 ### Related materialRead more...

• Less than, Equal to, Greater Than Symbols
• Order. Well-ordered sets
• Equivalence Relations
• Binary Relations
• Less, Equal, More
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