Answer
Not Reflexive
Symmetric
Not Transitive
Work Step by Step
Let X = {a, b, c} and P(X) be the power set of X.
A relation N is defined on P(X) as follows: For all
A, B ∈ P(X), A N B ⇔ the number of elements in A is
not equal to the number of elements in B.
Reflexive: for every A $\in$ P(X) ANA
it invalidates the relation since the number of elements in A is the same number of elements in A itself.
Symmetric: for every A, B $\in$ P(X) if ANB THEN BNA
let ANB be true, which means A doesn't have equal elements as B,
which also implies that B doesn't have equal elements as A,
then by definition of N BNA ( HENCE TRUE)
Transitive: If ANB and BNC then ANC for all A, B, C $\in$ P(X)
let A be a set with 8 elements, B with 5, C with 8.
which implies that
ANB and BNC
but ANC is false since A and C have 8 elements each.
which implies that the relation is not transitive.