Question

An “Anti-symmetric” relation need not be a reflexive relation: give an example.

Answer 1

Hint: First we learn the definition of Anti-symmetric relation and reflexive relation. Then we learn about the conditions for a relation to be Antisymmetric relation and reflexive relation. Then we solve the above question by giving an example.

Complete step-by-step answer:
We know that a relation R in mathematics is a connection between the elements of two or more sets.
Suppose we have a set A and if every element of set A maps for itself, then set A is a reflexive relation. It is written as $a\in A,$ and $(a,a)\in R$
For example- we have a set $A=\{1,2\}$ . For reflexive relation following condition must be satisfied- $R=\{(1,1),(2,2),(1,2),(2,1)\}$ .
Now let’s understand the anti-symmetric relation by an example.
A relation R is antisymmetric if $R(a,b)$ with $a\ne b$ then $R(b,a)$ must not hold.
For example- we have a relation $R=\{(1,1),(1,2)\}$ on the set $A=\{1,2\}$ . This relation is Antisymmetric relation because $(2,1)\notin R$
Also, the relation $R=\left\{(1,2)\right\}$ is Anti-symmetric relation but it is not reflexive as $(2,2)\notin R$ .
Hence it is proved that an “Anti-symmetric” relation need not be a reflexive relation.

Note: To solve this question students must have a clear idea about Anti-symmetric relation and reflexive relation. Reflexive relations can be symmetric relations. If we have anything other than the same ordered pair in a relation, it will not meet the Antisymmetric requirement.