Given , P and Q are equivalence relations..It is quite possible that the pair {x,y} exists in P and {y,z} exists in Q but {x,z} is not present in either of them..In that case union of P and Q will not contain the pair {x,z} , thereby violating the transitive property..Hence P union Q need not be equivalence relation as for being equivalence relation we need to the relation to be transitive..
For sake of convenience in understanding , we may take P = { (1,1) , (2,2) , (3,3) , (1,3) , (3,1) }
Q = { (1,1) , (2,2) , (3,3) , (3,2) , (2,3) }
So we can see that individually they are equivalence relations but on doing union the pair (1,3) and (3,2) is there due to which (1,2) should also be there..But it wont be included because (1,2) is present in neither P nor Q..
But there wont be such violation in P ∩ Q as if at all (x,y) is present in P and (y,z) is present in Q , then P ∩ Q will not include any of (x,y) or (y,z) and hence the transitivity is preserved.
Hence C) option should be correct.
NOTE : An empty set cannot generate non empty relation and hence the relation being empty is not reflexive and hence not an equivalence relation..However P and Q are given to be equivalence relations..So u have to take non empty set first and then non empty relation only then we can talk about P U Q or P ∩ Q .
