Prove that a point x is a limit point of a set A if and only if every e-neighborhood of x intersects A at some point other than x.

A relation R on a set A is defined to be irreflexive if, and only if, for every  x ∈ A,  x R x; asymmetric if, and only if, for every  x, y ∈ A  if  x R y  then y R x; intransitive if, and only if, for every  x, y,   z ∈ A,  if  x R y  and  y R z  then x R z. Let  A = {0, 1, 2, 3},  and define a relation  R4  on A as follows. R4 = (1, 2), (2, 1), (1, 3), (3, 1) Is  R4  irreflexive, asymmetric, intransitive, or none of these? R4 is irreflexive.R4 is asymmetric.R4 is intransitive.R4 is neither irreflexive, asymmetric, nor intransitive.

Let A, B ⊂R. Prove that (A ∩B)′ ⊂A′ ∩B′.(Recall that A′ is the set of limit points of A). b. Give an example of A and B for which (A ∩B)′ is a strict subset of A′ ∩B′.