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 (=)  


Not equal to (≠)  


The relation "not equal" is not reflexive: for no number a, a ≠ a. It is also not transitive, for example,
Less than (<)  


Being less means, in particular, not being equal to, so that this relation is not reflexive: it is not true that, say,
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
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Copyright © 19962018 Alexander Bogomolny