Question

Give an example of a set A and a binary relation R on A that is neither symmetric nor antisymmetric.

Answer 1

Let R be a binary relation defined on a set S.

Then, in the most basic terms, a symmetric relation is one in which for any ordered pair (x,y) in R, the ordered pair (y,x) must also be in R.

While an anti symmetric relation is one in which for any ordered pair (x,y) in R, the ordered pair (y,x) does NOT belong to R, unless x=y.

Example:

Now, consider a set A = {a,b,c}.

And R = {(a,b), (b,c), (c,b)}

Now,

• R is not symmetric as (a,b) belongs to R but (b,a) does not belong to R. And that violates the definition of a symmetric relation.
• R is not antisymmetric as for (b,c) belongs to R, we also have (c,b) in R. And so, also again the definition of an antisymmetric relation is being violated.