Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Inequality symbol"

m
m
 
Line 21: Line 21:
  
  
−
These symbols are also frequently used to represent the order [[relation]] in a [[partially ordered set]].  Note that in this more general setting, it is ''not'' necessarily true that <math>a \not > b \Longleftrightarrow a \leq b</math>, because it is also possible that <math>a</math> and <math>b</math> could be incomparable.
+
These symbols are also frequently used to represent the order [[relation]] in a [[partially ordered set]].  Note that in this more general setting, it is ''not'' necessarily true that <math>a \not > b \Longleftrightarrow a \leq b</math>, because it is also possible that <math>a</math> and <math>b</math> could be [[incomparable]].
 +
 
 +
[[Category:Notation]]

Latest revision as of 15:54, 9 February 2007

There are four symbols conventionally used to represent the notion of inequality.

If $a$ and $b$ are real numbers we write:

  • $a > b$ to mean that $a$ is strictly greater than $b$ (that is, $a$ cannot equal $b$).
  • $a \geq b$ to mean that $a$ is greater than or equal to (equivalently, "at least as large as") $b$.
  • $a < b$ to mean that $a$ is strictly less than $b$
  • $a \leq b$ to mean that $a$ is less than or equal to $b$.


We use a slash through an inequality symbol to represent that the given inequality does not hold. Thus for real numbers $a$ and $b$,

  • $a \not > b$ if and only if $a \leq b$
  • $a \not \geq b$ if and only if $a < b$
  • $a \not < b$ if and only if $a \geq b$
  • $a \not \leq b$ if and only if $a > b$
  • $\displaystyle a \neq b$ if and only if $a > b$ or $a < b$


These symbols are also frequently used to represent the order relation in a partially ordered set. Note that in this more general setting, it is not necessarily true that $a \not > b \Longleftrightarrow a \leq b$, because it is also possible that $a$ and $b$ could be incomparable.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Inequality_symbol&oldid=13020"