If two sets $A$ and $B$ have $99$ elements in common,then the number of elements common to each of the sets $A \times B$ and $B \times A$ is

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 $A \times \{B \cap \varnothing\} = \varnothing$.

If $P = \{a, b, c\}$ and $Q = \{r\}$,find the sets $P \times Q$ and $Q \times P$. Are these two products equal?

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?

Let $A = \{1, 2\}$,$B = \{1, 2, 3, 4\}$,$C = \{5, 6\}$,and $D = \{5, 6, 7, 8\}$. Verify that $A \times C$ is a subset of $B \times D$.

If $A = \{1, 2, 3, 4, \dots, 100\}$ and $B = \{51, 52, 53, \dots, 180\}$,then the number of elements in $(A \times B) \cap (B \times A)$ is: