Question

# 1.Consider a binary relation R on A. Recall that EA is the equality relation on A, that

is, E_{A} = {⟨a, a⟩ | a ∈ A}. Prove the following:

(a) R is reflexive if and only if E_{A} ⊆ R.

(b) R is symmetric if and only if R^{−1}= R.

(c) R is transitive if and only if R◦R⊆R.

(d) If R⊆R^{−1} then R^{−1} =R.

2.Consider a set A. Define a binary relation E on P(A) as follows:

⟨X,Y⟩∈E ⇐⇒ There exists a bijection f : X→Y

Prove that E is an equivalence relation on A. Also, if A is finite, how many equivalence classes with respect to E there are? Explain how you arrived at your answer.

#### Top Answer

The proofs and... View the full answer