Here you will learn what is antisymmetric relation on sets with definition and examples.
Let’s begin –
What is Antisymmetric Relation ?
Definition : Let A be any set. A relation R on set A is said to be an antisymmetric relation iff
(a, b) \(\in\) R and (b, a) \(\in\) R \(\implies\) a = b for all a, b \(\in\) A
(i) It follows from this definition that if (a, b) \(\in\) R but (b, a) \(\notin\) R, then also R is an antisymmetric relation.
(ii) The identity relation on a set A is an antisymmetric relation.
Given below are some antisymmetric relation examples.
Example : Let R be a relation on the set N of natural numbers defined by
x R y \(\iff\) ‘x divides y’ for all x, y \(\in\) N.
This relation is an antisymmetric relation on N. Since for any two numbers a, b \(\in\) N.
a | b and b | a \(\implies\) a = b i.e. a R b and b R a \(\implies\) a = b
It should be noted that this relation is not antisymmetric on the set Z of integers, because we find that for any non-zero integer a, a R (-a) and (-a) R a but a \(\ne\) – a.
Example : The relation \(\le\) (“less than or equal to”) on the set R of real numbers is antisymmetric, because a \(\le\) b and b \(\le\) a \(\implies\) a = b for all a, b \(\in\) R.