Give an example of a relation which is
symmetric but neither reflexive nor transitive.
Recall that for any binary relation R on set A. We have,
R is reflexive if for all x ∈ A, xRx.
R is symmetric if for all x, y ∈ A, if xRy, then yRx.
R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.
Let there be a set A.
A = {1, 2, 3, 4}
We need to define a relation which is symmetric but neither reflexive nor transitive.
Let there be a set A.
A = {1, 2, 3, 4}
Symmetric Relation:
{(1, 3), (3, 1)}
This is neither reflexive nor transitive.
∵ (1, 1) ∉ R
(3, 3) ∉ R
Hence, R is not reflexive.
∵ (1, 3) ∈ R and (3, 1) ∈ R
Then, (1, 1) ∉ R
Hence, R is not transitive.
Thus, the relation which is symmetric but neither nor transitive is:
R = {(1, 3), (3, 1)}