If $A = \{x : x \in \mathbb{N},\ x < 3\}$, find $A \times A \times A$ and the number of elements it contains.

If A = \x : x ∈ mathbbN,x < 3\, find A × A × A and – Class 11 Mathematics

Exercise: Cartesian Product · Q12

A = \{x : x \in \mathbb{N},\ x < 3\}

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Step 1: $A=\{x:x\in\mathbb{N},\,x<3\}=\{1,2\}$ , so $n(A)=2$ .

Step 2: $A\times A\times A$ consists of every ordered triple $(a,b,c)$ with $a,b,c\in A$ , and $n(A\times A\times A)=n(A)\times n(A)\times n(A)=2\times2\times2=8$ . …

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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 $A = \{x : x \in \mathbb{N},\ x < 3\}$ , find $A \times A \times A$ and the number of elements it contains.

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={x:x∈N, x<3}A = \{x : x \in \mathbb{N},\ x < 3\}A={x:x∈N, x<3}, find A×A×AA \times A \times AA×A×A and the number of elements it contains.

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Q.If $A = \{x : x \in \mathbb{N},\ x < 3\}$ , find $A \times A \times A$ and the number of elements it contains.

Unlock everything free for 14 days