Thread: antisymmetric

R = {(a,b) is an element of A cross B with (b,a) an element of R inverse}

Proof:
If R is antisymmetric, then R inverse is antisymmetric

-if you can not prove this I must show a counterexample thanks

You need to show if R is antisymettric, meaning if a<=b and b<=a implies a=b. Then $\displaystyle R^{-1}$ has the same property. It should be clear how to do this by definition.

If $\displaystyle (x,y) \in R^{ - 1} \wedge (y,x) \in R^{ - 1} \quad \Rightarrow \quad (y,x) \in R \wedge (x,y) \in R.$
But we know that $\displaystyle R$ is antisymmetric so $\displaystyle x=y$.