Set Relation Exercises

Suppose that A and B are sets with relations ~_A and ~_B, respectively. Suppose further that f: A → B is a function with the property that x ~_A y if and only if f(x) ~_B f(y).

• Prove that if ~_B is reflexive, then ~_A is reflexive.
• Give a counterexample showing that the converse is false.
• Prove that if ~_A is reflexive, then the restriction of ~_B to f(A) is reflexive.
• Prove that if ~_B is reflexive and f is not injective, then ~_A is not antisymmetric.