If A = {1, 2, 3, 4}, define relations on A which have properties of being
reflexive, symmetric and 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.
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, symmetric and transitive.
The relation must be defined on A.
Reflexive Relation:
R = {(1, 1), (2, 2), (3, 3), (4, 4)}
Or simply shorten it and write,
R = {(1, 1), (2, 2)} …(1)
Symmetric Relation:
R = {(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3)}
Or simply shorten it and write,
R = {(1, 2), (2, 1)} …(2)
Combine results (1) and (2), we get
R = {(1, 1), (2, 2), (1, 2), (2, 1)}
It is reflexive, symmetric as well as transitive as per definition.
Similarly, we can find other combinations too.
Thus, we have got the relation which is reflexive, symmetric as well as transitive.