The set $A = \{ x : x \ne x \}$ represents:

Let $P$ and $T$ be the subsets of the $xy$-plane defined by $P = \{(x, y) : x > 0, y > 0 \text{ and } x^2 + y^2 = 1\}$ and $T = \{(x, y) : x > 0, y > 0 \text{ and } x^8 + y^8 < 1\}$. Then,$P \cap T$ is

For any sets $A$ and $B$,show that $P(A \cap B) = P(A) \cap P(B).$

If $Q = \{x : x = \frac{1}{y}, \text{ where } y \in N\}$,then which of the following is true?

In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give an example.
If $A \subset B$ and $x \notin B,$ then $x \notin A$.

Write the set builder form of the set $ A = \{-1, 1\} $.