Formally prove the following statements about a relation R ⊆ A × A, using the definitions of the given properties. 1 Prove that, if R is irreflexive and transitive, then R is asymmetric. 2 Prove...

Formally prove the following statements about a relation R ⊆ A × A, using the definitions of the given properties.

1 Prove that, if R is irreflexive and transitive, then R is asymmetric.

2 Prove Theorem 8.2: show that R is transitive if and only if R ◦ R ⊆ R.

3 Theorem 8.2 cannot be stated with an = instead of ⊆ (although I actually made this mistake in a

previous draft!). Give an example of a transitive relation R where R ◦ R ⊂ R (that is, where R ◦ R 6= R).