If $A=\{2,\ 3\}$, $B=\{4,\ 5\}$ and $C=\{5,\ 6\}$ then the $A imes(B\cap C)$ is
(A) $\{(3,\ 5),\ (2,\ 6)\}$
(B) $\{(2,\ 4),\ (5,\ 3)\}$
(C) $\{(5,\ 2),\ (5,\ 3)\}$
(D) $\{(2,\ 5),\ (3,\ 5)\}$

Question

If $A=\{2,\ 3\}$, $B=\{4,\ 5\}$ and $C=\{5,\ 6\}$ then the $A	imes(B\cap C)$ is
(A) $\{(3,\ 5),\ (2,\ 6)\}$
(B) $\{(2,\ 4),\ (5,\ 3)\}$
(C) $\{(5,\ 2),\ (5,\ 3)\}$
(D) $\{(2,\ 5),\ (3,\ 5)\}$

Accepted Answer

> [!TLDR]
> $B\cap C=\{5\}$, so $A	imes\{5\}=\{(2,5),(3,5)\}$.

### Step 1 — Intersection $B\cap C$

Keep only the elements common to both sets:

$$B\cap C=\{4,5\}\cap\{5,6\}=\{5\}.$$

### Step 2 — Cartesian product $A	imes(B\cap C)$

Pair each element of $A=\{2,3\}$ with each element of $\{5\}$, writing each ordered pair as (element from $A$, element from $B\cap C$):

$$A	imes\{5\}=\{(2,5),\ (3,5)\}.$$ …

Cartesian Product of Sets

Step 1 — Intersection $B\cap C$

Step 2 — Cartesian product $A\times(B\cap C)$

Step 1 — Intersection B∩CB\cap CB∩C

Step 2 — Cartesian product A×(B∩C)A\times(B\cap C)A×(B∩C)

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Step 1 — Intersection $B\cap C$

Step 2 — Cartesian product $A\times(B\cap C)$

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