Mark each statement True or False. Justify each answer. (a) If p is a partition of set S, then we can obtain a relation R on S by defining x R y iff x and y are in the same piece of the partition. (b)...

Mark each statement True or False. Justify each answer.

(a) If p is a partition of set S, then we can obtain a relation R on S by defining x R y iff x and y are in the same piece of the partition.

(b) In any relation R on a set S, we always have x R x for all x ∈ S.

(c) If R is a relation on S, then { y ∈ S: y R x} determines a partition of S.

(d) If p is a partition of S and x ∈ S, then x ∈ A for some A ∈ p