Question

a) there exists a non-empty relation which is both symmetric and anti-symmetric.

Answer 1

Answer:

17

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:

Step-by-step explanation: