How many players must be at least a set A so that it can define a relation R which is neither symmetric nor antisymmetric. Answer detailed explanation.

- Nov 12th 2010, 01:23 PMametfDefine a relation R which is neither symmetric nor antisymmetric
How many players must be at least a set A so that it can define a relation R which is neither symmetric nor antisymmetric. Answer detailed explanation.

- Nov 12th 2010, 01:52 PMemakarov
Do they refer to constituent of sets as "players" instead of "elements" now? :)

Try various relations on sets with 0, 1, 2, and 3 elements. Consider, for example, antisymmetry: for all x and y, if R(x,y) and R(y,x), then x = y. Note that if it is not the case that R(x,y) and R(y,x) for some x, y, this fact does not violate antisymmetry because an implication is vacuously true when the assumption is false. So, to break antisymmetry, you need to have x and y such that the assumption is true but the conclusion is false. - Nov 13th 2010, 07:27 AMPlato
Consider the set $\displaystyle \{a,b,c\}$ and the relation $\displaystyle \math{K}=\{(a,b),(b,c),(b,a)\}$.

Is true that $\displaystyle \math{K}$ is neither antisymmetric or symmetric?

Can you improve on the number three?