$A = \{a, b, p\}$, $B = \{2, 3\}$, $C = \{p, q, r, s\}$ then $n[(A \cup C) imes B]$ is :
(A) 8
(B) 20
(C) 12
(D) 16

Question

$A = \{a, b, p\}$, $B = \{2, 3\}$, $C = \{p, q, r, s\}$ then $n[(A \cup C) 	imes B]$ is :
(A) 8
(B) 20
(C) 12
(D) 16

Accepted Answer

> [!TLDR]
> $n(A \cup C) = 6$, $n(B) = 2$, so $n[(A \cup C) 	imes B] = 6 	imes 2 = 12$.

### Step 1 — Form the union
$$A \cup C = \{a, b, p\} \cup \{p, q, r, s\} = \{a, b, p, q, r, s\}$$
The element $p$ is common and counted only once, so $n(A \cup C) = 6$.

### Step 2 — Note the size of $B$
$$B = \{2, 3\} \;\Rightarrow\; n(B) = 2$$

### Step 3 — Cardinality of the Cartesian product
> [!FORMULA]
> $$n(X 	imes Y) = n(X) 	imes n(Y) = 6 	imes 2 = 12$$ …

Step 1 — Form the union

Step 2 — Note the size of BBB

Step 3 — Cardinality of the Cartesian product

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Step 2 — Note the size of $B$

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