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".


Equal to (=)

Reflexivity:a = a
Symmetry:a = b ⇒ b = a
Transitivity:a = b, b = c ⇒ a = c


Not equal to (≠)

Antireflexivity:not a ≠ a
Symmetry:a ≠ b ⇒ b ≠ a


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 (<)

Antireflexivity:not a < a
Asymmetry:a < b ⇒ not b < a
Transitivity:a < b, b < c ⇒ a < c


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 (>)

Antireflexivity:not a > a
Asymmetry:a > b ⇒ not b > a
Transitivity:a > b, b > c ⇒ a > c


Less than or equal to (≤)

Reflexivity:a ≤ a
Antisymmetry:a ≤ b, b ≤ a ⇒ a = b
Transitivity:a ≤ b, b ≤ c ⇒ a ≤ c


And also


Greater than or equal to (≥)

Reflexivity:a ≥ a
Antisymmetry:a ≥ b, b ≥ a ⇒ a = b
Transitivity:a ≥ b, b ≥ c ⇒ a ≥ c


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 material
Read more...

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

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