Consider a binary relationRonA. Recall thatEAis the equality relation onA, that is,E A ={a, a |aA}. Prove the following: (a)Ris reflexive if and only...
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1.Consider a binary relation R on A. Recall that EA is the equality relation on A, that

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

(a) R is reflexive if and only if EA ⊆ 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.

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The proofs and... View the full answer

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Problem 2.
We consider a set A. A binary relation E on P(A) be defined as follows:
(X, Y) EE <> There exists a bijection f : X - Y
Let X E P(A)
We know that the identity map ix defined by :...

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