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 $ \left( a,a \right)\in R $
For example- we have a set $ A=\left\{ 1,2 \right\} $ . For reflexive relation following condition must be satisfied- $ R=\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 1,2 \right),\left( 2,1 \right) \right\} $ .
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=\left\{ \left( 1,1 \right),\left( 1,2 \right) \right\} $ on the set $ A=\left\{ 1,2 \right\} $ . This relation is Antisymmetric relation because $ \left( 2,1 \right)\notin R $
Also, the relation $ R=\left\{ \left( 1,2 \right) \right\} $ is Anti-symmetric relation but it is not reflexive as $ \left( 2,2 \right)\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.