Give an example of a relation which is
reflexive and symmetric but not 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 on A which is reflexive and symmetric but not transitive.
Let there be a set A.
A = {1, 2, 3, 4}
Reflexive relation:
R = {(1, 1), (2, 2), (3, 3), (4, 4)} …(1)
Symmetric relation:
R = {(3, 4), (4, 3)} …(2)
Combine results (1) and (2), we get
R = {(1, 1), (2, 2), (3, 3), (4, 4), (3, 4), (4, 3)}
Check for Transitivity:
If (3, 4) ∈ R and (4, 3) ∈ R
Then, (3, 3) ∈ R
∀ 3, 4 ∈ A [∵ A = {1, 2, 3, 4}]
So eliminate (3, 3) from R, we get
R = {(1, 1), (2, 2), (4, 4), (3, 4), (4, 3)}
Check for Transitivity:
If (4, 3) ∈ R and (3, 4) ∈ R
Then, (4, 4) ∈ R
∀ 3, 4 ∈ A
So, eliminate (4, 4) from R, we get
R = {(1, 1), (2, 2), (3, 4), (4, 3)}
Thus, the relation which is reflexive and symmetric but not transitive is:
R = {(1, 1), (2, 2), (3, 4), (4, 3)}