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.