Prove Theorem 8.1, connecting the symmetry/asymmetry/antisymmetry of a relation R to the inverse R−1 of R. 1. Prove that R is symmetric if and only if R ∩ R −1 = R = R −1 . 2. Prove that R is...

Prove Theorem 8.1, connecting the symmetry/asymmetry/antisymmetry of a relation R to the inverse R−1

of R.

1. Prove that R is symmetric if and only if R ∩ R

−1 = R = R⁻¹.

2. Prove that R is antisymmetric if and only if R ∩ R

−1 ⊆ {ha, ai : a ∈ A}.

3. Prove that R is asymmetric if and only if R ∩ R

−1 = ∅.