Question

can a binary relation be both symmetric and antisymmetric? justify ​

Answer 1

Answer:

A relation can be both symmetric and antisymmetric, for example the relation of equality. It is symmetric since a=b⟹b=a but it is also antisymmetric because you have both a=b and b=a if a=b.

A convenient way of thinking about these properties is from a graph-theoretical perspective.

Let us define a graph (technically a directed multigraph with no parallel edges) in the following way:

Have a vertex for every element of the set. Draw an edge with an arrow from a vertex a to a vertex b iff there a is related to b (i.e. aRb, or equivalently (a,b)∈R).

If an element is related to itself, draw a loop, and if a is related to b and b is related to a, instead of drawing a parallel edge, reuse the previous edge and just make the arrow double sided (↔)

For example, for the set {1,2,3} the relation R={(1,1),(1,2),(2,3),(3,2)} has the following graph:

enter image description here

Definitions:

SymmetricAnti-Symmetricset theoreticalIf aRb then bRaIf aRb and bRa then a=bgraph theoreticalAll arrows (not loops) are double sidedAll arrows (not loops) are single sided

You see then that if there are any edges (not loops) they cannot simultaneously be double-sided and single-sided, but loops don't matter for either definition. Any relation on a set A that is both anti-symmetric and symmetric then has its graph consisting of only loops (i.e. is of the form R={(a,a) | a∈S⊆A} for some S⊆A.

Any relation whose graph contains both types of arrows (single-sided and doublesided) will be neither symmetric nor antisymmetric.