If $X = \{p, q, r, s\}$ and $Y = \{1, 2\}$, find $n(X)$, $n(Y)$ and $n(X \times Y)$. Also list $X \times Y$.

If X = \p, q, r, s\ and Y = \1, 2\, find n(X), n(Y) and n(X × Y). Also list X × Y.

Exercises · Q10

X = \{p, q, r, s\}

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Step 1 — Count each set. $X = \{p,q,r,s\}$ has 4 elements, so $n(X) = 4$ . $Y = \{1,2\}$ has 2 elements, so $n(Y) = 2$ .

Step 2 — Apply the cardinality-of-product formula. $n(X \times Y) = n(X) \times n(Y) = 4 \times 2 = 8$ .

Step 3 — Verify by direct listing. Pairing each element of $X$ with each element of $Y$ :

With $p$ : $(p,1), (p,2)$
With $q$ : $(q,1), (q,2)$
With $r$ : $(r,1), (r,2)$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CA Foundation 2026Set jan-20261 markMCQ
$A=\{2,\ 3\}$
›Reveal solutionSolution

$B\cap C=\{5\}$ , so $A\times\{5\}=\{(2,5),(3,5)\}$ .

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 $A\times(B\cap C)$

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

$A\times\{5\}=\{(2,5),\ (3,5)\}.$ …
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CA Foundation 2025Set jan-20251 markMCQ
$A = \{a, b, p\}$
›Reveal solutionSolution

$n(A \cup C) = 6$ , $n(B) = 2$ , so $n[(A \cup C) \times B] = 6 \times 2 = 12$ .

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
$n(X \times Y) = n(X) \times n(Y) = 6 \times 2 = 12$
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CA Foundation 2023Set jun-20231 markMCQ
$A=\{a, b, c\}, B=\{b, c, d\}$
›Reveal solutionSolution

A − B = {a}, B ∩ C = {c, d}, so {a} × {c, d} = {(a, c), (a, d)}.

Step 1 — Set difference

A = {a, b, c}, B = {b, c, d}: elements of A not in B ⇒ A − B = {a}.

Step 2 — Intersection

B = {b, c, d}, C = {a, d, c}: common elements ⇒ B ∩ C = {c, d}.

Step 3 — Cartesian product
$\{a\} \times \{c, d\} = \{(a, c), (a, d)\}$
Watch out

Ordered pairs: the first coordinate must come from A − B, so (c, a) and (d, a) are wrong. …

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Q.If $X = \{p, q, r, s\}$ and $Y = \{1, 2\}$ , find $n(X)$ , $n(Y)$ and $n(X \times Y)$ . Also list $X \times Y$ .

Concept understanding — Cartesian Product of Sets

Step 1 — Intersection $B\cap C$

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

Step 1 — Form the union

Step 2 — Note the size of $B$

Step 3 — Cardinality of the Cartesian product

Step 1 — Set difference

Step 2 — Intersection

Step 3 — Cartesian product

Q.If X={p,q,r,s}X = \{p, q, r, s\}X={p,q,r,s} and Y={1,2}Y = \{1, 2\}Y={1,2}, find n(X)n(X)n(X), n(Y)n(Y)n(Y) and n(X×Y)n(X \times Y)n(X×Y). Also list X×YX \times YX×Y.

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Q.If $X = \{p, q, r, s\}$ and $Y = \{1, 2\}$ , find $n(X)$ , $n(Y)$ and $n(X \times Y)$ . Also list $X \times Y$ .

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