Let $A$ and $B$ be sets with $|A| = 5$ and $|B| = 7$. Let $n = |A \times B|$.

Question

Let $A$ and $B$ be sets with $|A| = 5$ and $|B| = 7$. Let $n = |A \times B|$.

A. $n \ge 7$

B. $n = 12$

C. $n \le 35$

D. $n = 35$

I thought the answer should be D, but my friend said the correct one is C, so I don't really get it. Is it C or D and why?

Thank You.

Answer 1

The rule of product states that for sets $A$ and $B$ one has

$|A\times B|=|A|\times |B|$

In this case with $|A|=5$ and $|B|=7$ you have $|A\times B|=|A|\times |B|=5\times 7 = 35$ so you specifically have $n=35$

Since $35\geq 7$ you have $a)$ is true.

Since $35\neq 12$ you have $b)$ is false.

Since $35\leq 35$ you have $c)$ is true.

Since $35=35$ you have $d)$ is true.