Mathematics

Cartesian product of sets

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In Class 11 Mathematics, under Relations and Functions, students learn how to form the Cartesian product of two sets as the set of all possible ordered pairs (a, b), where a belongs to the first set and b to the second. The topic explains why the order of elements matters, how to represent products using roster form and diagrams, and how to find their number of elements. This foundation helps students describe relations as subsets of a Cartesian product and understand domain and codomain.

Practice questions

01 If (A={1,2,3,4}) and (B={5,6,7}), how many subsets of (A\times B) contain ((1,5)) and ((4,7)) but do not contain ((2,6))?

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Answer and explanation

Cartesian product of sets

Practice questions

01 If (A={1,2,3,4}) and (B={5,6,7}), how many subsets of (A\times B) contain ((1,5)) and ((4,7)) but do not contain ((2,6))?

02 If (|A|=2) and (|B|=5), what is the number of subsets of (A\times B) having exactly (5) elements?

03 If (A={2,3,4,5,6,7,8}), (B={1,2,3,4,5}), and (R={(a,b):a+3b=17}), what is (|R|)?

04 If (A={0,1,2,3,4,5}) and (B={0,1,2,3,4,5}), how many pairs ((a,b)) in (A\times B) satisfy (a^2+b^2=25)?

05 If (A={1,2,3,4,5,6}) and (B={1,2,3,4,5,6}), how many pairs ((a,b)) in (A\times B) satisfy both (a\mid b) and (a<b)?

06 If \(A=\{1,2,3,4,5\}\) and \(B=\{2,3,4,5,6,7\}\), how many ordered pairs \((a,b)\) in \(A\times B\) satisfy both conditions: \(a+b\) is prime and \(a<b\)?

07 If (A={1,2,3,4}), (B={1,2,3,4}), and (S={(a,b):a+b\text{ is even}}), how many pairs are in the complement of (S) relative to (A\times B)?

08 If (A={0,1,2,3}), (B={0,1,2,3,4}), and (R={(a,b):b\ge a^2-1}), what is (|R|)?

09 If (A={1,2,3,4,5}) and (B={1,2,3,4,5}), how many pairs ((a,b)) in (A\times B) satisfy (\gcd(a,b)=1) and (a+b) even?

10 If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(C=\{2,3,5\}\), what is the value of \(\lvert A\times(B\setminus C)\rvert+\lvert(A\times B)\cap(A\times C)\rvert\)?

11 If (A={1,2,3,4}) and (B={1,2,3,4,5,6}), how many pairs ((a,b)) in (A\times B) satisfy (ab\le12) and (a+b) is odd?

12 If (A={1,2,3,4}) and (B={2,3,5,7,11}), how many pairs ((a,b)) in (A\times B) have (b-a) prime?

13 If \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\), and \(C=\{0,1\}\), how many ordered triples \((a,b,c)\) in \(A\times B\times C\) satisfy \(a=b+c\)?

14 If (A={1,2,3,4,5,6}) and (B={1,2,3,4,5,6}), how many pairs ((a,b)) in (A\times B) satisfy (\operatorname{lcm}(a,b)=6)?

15 If A={1,2,3,4}, B={1,2,3,4}, and U=A×B, how many ordered pairs (a,b) in U satisfy a<b or a+b=5?

16 If (A={0,1,2,3,4}) and (B={0,1,2,3,4}), how many pairs ((a,b)) in (A\times B) satisfy (a+b\le4) and (a\ne b)?

17 If (A={1,2,3,4,5,6}) and (B={1,2,3,4,5,6}), how many pairs ((a,b)) in (A\times B) have (a+b) divisible by (3) and (a\ne b)?

18 If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5,6\}\), how many ordered pairs \((a,b)\) in \(A\times B\) satisfy that \(b\) is a multiple of \(a\) and \(b\ne a\)?

19 If (A={1,2,3,4}) and (B={0,1,2,3,4,5}), how many pairs ((a,b)) in (A\times B) satisfy (b\equiv a^2\pmod{5})?

20 If (A={1,2,3,4}), (B={1,2,3,4}), and (R={(a,b):a+b\text{ is odd}}), what is the number of subsets of (R)?

21 If (A={1,2,3,4,5}) and (B={1,2,3,4,5}), how many pairs ((a,b)) in (A\times B) satisfy (a<b) and (a+b) is even?

22 If (A={1,2,3,4,5,6}) and (B={1,2,3,4,5,6}), how many pairs ((a,b)) in (A\times B) have both (a) and (b) prime or both composite?

23 If (A={1,2,3,4,5}), (B={0,1,2,3,4}), and (R={(a,b):a-b\in{1,3}}), what is (|R|)?

24 If (A={1,2,3,4}), (B={1,2,3,4}), and (R=A\times B), what is the minimum number of pairs to remove from (R) so that no pair ((a,a)) remains?

25 If (A={1,2,3}), (B={1,2,3}), and (C={1,2,3}), how many triples ((a,b,c)) in (A\times B\times C) satisfy (a+b+c=6)?

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