Question

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Answer 1

$\text{\bf{reflexive relation}}$ A relation is said to be reflexive relation, if every element of A is related to itself.
thus,(a,a) ∈ R , for all a∈ A => R is reflexive

$\text{\bf{symmetric relation}}$ A relation is said to be symmetric relation, if
(a, b) ∈ R => (b, a) ∈ R , for all a, b ∈ A
e.g., aRb => bRa, for all a, b ∈ A => R is symmetric.

$\text{\bf{transitive relation}}$ A relation is said to be transitive relation if
(a, b) ∈ R and (b, c) ∈ R => (a,c) ∈ R for all a,b,c ∈ A

now, let's check all relations :
Let us take A = {1, 2, 3}
A relation R is defined on set A as R = {(1,2), (2,1)}
It is seen that (1,1), (2,2), (3,3) ∉ R.
Therefore, R is not reflexive.

Now, we can see that (1,2) ∈ R and (2,1) ∈ R
Therefore, R is symmetric.
And now, (1,2), (2,1) ∈ R
But, (1,1) ∉ R.Therefore, R is not transitive.

Therefore, R is symmetric but neither reflexive, nor transitive.