If $A = \{1, 2\}$ and $B = \{3, 4, 5\}$, find $A \times B$ and $|A \times B|$.

If A = \1, 2\ and B = \3, 4, 5\, find A × B and |A × B|.

Worked Examples · Example 4

A = \{1, 2\}

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Step 1 — Recall the definition. $A \times B = \{(a,b) : a \in A, b \in B\}$ , pairing every element of $A$ (first component) with every element of $B$ (second component).

Step 2 — List systematically, taking each element of $A$ in turn.

With $a = 1$ : $(1,3), (1,4), (1,5)$
With $a = 2$ : $(2,3), (2,4), (2,5)$

Step 3 — Combine. $A \times B = \{(1,3), (1,4), (1,5), (2,3), (2,4), (2,5)\}$ — 6 pairs in total. …

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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 $A = \{1, 2\}$ and $B = \{3, 4, 5\}$ , find $A \times B$ and $|A \times B|$ .

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 A={1,2}A = \{1, 2\}A={1,2} and B={3,4,5}B = \{3, 4, 5\}B={3,4,5}, find A×BA \times BA×B and ∣A×B∣|A \times B|∣A×B∣.

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Q.If $A = \{1, 2\}$ and $B = \{3, 4, 5\}$ , find $A \times B$ and $|A \times B|$ .

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