If $A = \{a, b\}$ and $B = \{1, 2, 3\}$, find $A \times B$ and $B \times A$. Are the two sets equal?

If A = \a, b\ and B = \1, 2, 3\, find A × B and B × A. Are the two sets equal?

Exercise: Cartesian Product · Q9

A = \{a, b\}

West Bengal WbchseTextbookSubjectiveImportance★★★★★

39% · 12/31 Questions

✓ Free question

Step 1: $A\times B$ pairs each element of $A=\{a,b\}$ with every element of $B=\{1,2,3\}$ : $A\times B=\{(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)\}$ .

Step 2: $B\times A$ pairs each element of $B$ with every element of $A$ : $B\times A=\{(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)\}$ .

Step 3: Comparing, e.g. $(a,1)\in A\times B$ but $(a,1)\notin B\times A$ (since in $B\times A$ the first component must come from $B$ ); so $A\times B\ne B\times A$ , even though both have $6$ elements.

✓Final answer

A\times B=\{(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)\}

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

Order matters in an ordered pair. The first coordinate must come from $A$ and the second from $B\cap C$ — a pair like $(5,2)$ (option C) reverses this and is wrong. Also compute the intersection before the product, not after.

Tip

When one of the sets in a Cartesian product has just a single element, the product simply "attaches" that element to each member of the other set — no need to enumerate a full grid.

✓Final answer
$\{(2,\ 5),\ (3,\ 5)\}$
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
$n(X \times Y) = n(X) \times n(Y) = 6 \times 2 = 12$
Why the other options are wrong: (D) 16 counts $p$ twice ( $7 \times 2$ -ish slip) or mis-sizes a set; (A) 8 uses $n(A)=4$ ?; (B) 20 mishandles the union.

Watch out

In the union, count the shared element $p$ only ONCE — $n(A\cup C) = 6$ , not 7.

Tip

$n(A \cup C) = n(A) + n(C) - n(A \cap C) = 3 + 4 - 1 = 6$ if you prefer the inclusion–exclusion route.

✓Final answer
$(C) 12$
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.

Tip

|A − B| × |B ∩ C| = 1 × 2 = 2 pairs — a quick count of how many pairs to expect.

✓Final answer
$\{(a, c), (a, d)\}$

Note

This answer is verified — independently worked and reviewed by our experienced subject lecturers before we published it.

←Previous questionExample 8 — If f(x) = x² + 1 and g(x) = 2x - 3, find (f+g)(x), (f-g)(x), (fg)(x) a…→Next questionQ10 — If n(A) = 5 and n(B) = 3, find n(A × B) and n(B × A).

Q.If $A = \{a, b\}$ and $B = \{1, 2, 3\}$ , find $A \times B$ and $B \times A$ . Are the two sets equal?

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={a,b}A = \{a, b\}A={a,b} and B={1,2,3}B = \{1, 2, 3\}B={1,2,3}, find A×BA \times BA×B and B×AB \times AB×A. Are the two sets equal?

🎓Unlock everything free for 14 days

Q.If $A = \{a, b\}$ and $B = \{1, 2, 3\}$ , find $A \times B$ and $B \times A$ . Are the two sets equal?

Unlock everything free for 14 days