If A = {1, 2, 3, 4}, define relations on A which have properties of being
reflexive, transitive but not 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.
Using these properties, we can define R on A.
A = {1, 2, 3, 4}
We need to define a relation (say, R) which is reflexive, transitive but not symmetric.
Let us try to form a small relation step by step.
The relation must be defined on A.
Reflexive relation:
R = {(1, 1), (2, 2), (3, 3), (4, 4)} …(1)
Transitive relation:
R = {(1, 2), (2, 1), (1, 1)}, is transitive but also symmetric.
So, let us define another relation.
R = {(1, 3), (3, 2), (1, 2)}, is transitive and not symmetric. …(2)
Let us combine (i) and (ii) relation.
R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 2), (1, 2)} …(A)
(A) can be shortened by eliminating (3, 2) and (1, 2) from R.
R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3)} …(B)
Further (B) can be shortened by eliminating (2, 2) and (4, 4).
R = {(1, 1), (3, 3), (1, 3)} …(C)
All the results (A), (B) and (C) is correct.
Thus, we have got the relation which is reflexive, transitive but not symmetric.