If $A \times B = \{(a, x), (a, y), (b, x), (b, y)\}$,find $A$ and $B$.

If $A = \{1, 2, 3, 4\}$ and $B = \{a, b\}$,then the Cartesian product $A \times B$ is:

If $A = \{1, 2, 3\}$ and $B = \{3, 8\}$,then $(A \cup B) \times (A \cap B)$ is

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 $y$.

State whether the following statement is true or false. If the statement is false,rewrite it correctly.
If $A = \{1, 2\}$ and $B = \{3, 4\}$,then $A \times \{B \cap \varnothing\} = \varnothing$.

Let $A = \{a, b, c\}$ and $B = \{1, 2\}$. Consider a relation $R$ defined from set $A$ to set $B$. Then $R$ is a subset of which of the following sets?