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 | ||
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≠ | 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 (=) | |||||||||
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Not equal to (≠) | ||||||
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The relation "not equal" is not reflexive: for no number a, a ≠ a. It is also not transitive, for example,
Less than (<) | |||||||||
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Being less means, in particular, not being equal to, so that this relation is not reflexive: it is not true that, say,
Greater than (>) | |||||||||
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Less than or equal to (≤) | |||||||||
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And also
Greater than or equal to (≥) | |||||||||
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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
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Copyright © 1996-2018 Alexander Bogomolny71534825