If A = {1, 2, 3, 4}, define relations on A which have properties of being
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.
Using these properties, we can define R on A.
A = {1, 2, 3, 4}
We need to define a relation (say, R) which is symmetric but neither reflexive nor transitive.
The relation R must be defined on A.
Symmetric relation:
R = {(1, 2), (2, 1)}
Note that, the relation R here is neither reflexive nor transitive, and it is the shortest relation that can be form.
Similarly, we can also write:
R = {(1, 3), (3, 1)}
Or R = {(3, 4), (4, 3)}
Or R = {(2, 3), (3, 2), (1, 4), (4, 1)}
And so on…
All of these are right answers.
Thus, we have got the relation which is symmetric but neither reflexive nor transitive.