Give an example of a relation which is
transitive but neither reflexive nor symmetric.
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 transitive but neither reflexive nor symmetric.
Let there be a set A.
A = {1, 2, 3}
Transitive Relation:
R = {(2, 4), (4, 1), (2, 1)}
This is neither reflexive nor symmetric.
∵ (1, 1) ∉ R
(2, 2) ∉ R
(4, 4) ∉ R
Hence, R is not reflexive.
∵ if (2, 4) ∈ R
Then, (4, 2) ∉ R
Hence, R is not symmetric.
Thus, the relation which is transitive but neither reflexive nor symmetric is:
R = {(2, 4), (4, 1), (2, 1)}