How to determine equivalence relation on a set of ordered pair

Question

I'm new to discrete maths and struggling with Relations, need to test the following relations for equivalence on the set X also find an equivalence class if it is equivalence relation.

Ques 1. X = R, x ∼ y <=> 2y < x² + 1

• For reflexivity

x ∼ x <=> 2x < x² + 1

for x = 1, the statement is false.

=> It is not reflexive.

• For symmetry[updated]

x ∼ y <=> 2y < x² + 1

y ∼ x <=> 2x < y² + 1

for x = 1 and y = 5 in 2y < x² + 1

10 < 2, which is false

=> It is not symmetric.

• For transitivity[updated]

x ∼ y <=> 2y < x² + 1

y ∼ z <=> 2z < y² + 1

x ∼ z <=> 2y < x² + 1

similar to symmetry

=> It is not transitive.

Ques 2. X = 2^N, A ∼ B <=> (A ∪ B)' = N

I'm not sure how to interpret X = 2^N and check if it is an equivalence relation.

Answer 1

For 2, $X$ is the set of subsets of $\{1,2,3,\ldots,N\}$. $A$ and $B$ are two of those subsets. They are related if the complement of their union is all of $\{1,2,3,\ldots,N\}$. First use DeMorgan's law on the complement of the union to get ??? Then check RST.