If $\left(\frac{x}{3}+1, y-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right),$ find the values of $x$ and $y$

The Cartesian product $A$ $\times$ $A$ has $9$ elements among which are found $(-1,0)$ and $(0,1).$ Find the set $A$ and the remaining elements of $A \times A$.

Let $A$ and $B$ be two sets such that $n(A)=3$ and $n(B)=2 .$ If $(x, 1),(y, 2),(z, 1)$ are in $A \times B$, find $A$ and $B$, where $x, y$ and $z$ are distinct elements.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

If $P=\{m, n\}$ and $Q=\{n, m\},$ then $P \times Q=\{(m, n),(n, m)\}.$

If $A=\{-1,1\},$ find $A \times A \times A.$