If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.
Hint. Two ordered pairs are equal only when their first elements match and their second elements match, separately.
Equate first elements: x/3 + 1 = 5/3, so x/3 = 5/3 - 3/3 = 2/3, giving x = 2. Equate second elements: y - 2/3 = 1/3, so y = 1/3 + 2/3 = 1.
✦ Working through each part gives: x = 2, y = 1.
If the set A has 3 elements and B = {3, 4, 5}, find the number of elements in A x B.
Hint. The size of a Cartesian product is simply the product of the two set sizes.
n(A) = 3 and n(B) = 3, so n(A x B) = n(A) x n(B) = 3 x 3 = 9.
✦ Working through each part gives: 9 elements.
If G = {7, 8} and H = {5, 4, 2}, find G x H and H x G.
Hint. Pair every element of the first set with every element of the second, keeping the stated order.
G x H pairs each of 7, 8 with each of 5, 4, 2 in that order. H x G pairs each of 5, 4, 2 with each of 7, 8 in that order -- a different, though equally sized, set of pairs.
✦ Working through each part gives: g x H = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}. H x G = {(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)}.
State whether true or false, correcting any false statement: (i) If P={m,n} and Q={n,m}, then PxQ={(m,n),(n,m)}. (ii) If A and B are non-empty sets, A x B is a non-empty set of ordered pairs (x,y) with x in A, y in B. (iii) If A={1,2}, B={3,4}, then A x (B intersect empty set) = empty set.
Hint. Write out the full Cartesian product for (i) rather than guessing which pairs belong -- Q has the same two elements as P, just listed in a different order.
(i) Q={n,m} is the same set as {m,n}, so P x Q actually has FOUR pairs: (m,m),(m,n),(n,m),(n,n) -- the statement lists only two, so it is false. (ii) this is exactly the definition of a Cartesian product of non-empty sets, so it is true. (iii) B intersect empty set is empty, and any set times the empty set is empty, so this is true.
✦ Working through each part gives: (i) False -- correctly, PxQ={(m,m),(m,n),(n,m),(n,n)}. (ii) True. (iii) True.
If A = {-1, 1}, find A x A x A.
Hint. Each of the three positions in the triple can independently be -1 or 1, giving 2x2x2 combinations.
List every combination of -1 and 1 across three positions systematically.
✦ Working through each part gives: axAxA = {(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),(1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)}, 8 triples in all.
If A x B = {(a,x),(a,y),(b,x),(b,y)}, find A and B.
Hint. A is exactly the set of first elements appearing across all the pairs; B is exactly the set of second elements.
The distinct first elements appearing are a and b; the distinct second elements appearing are x and y.
✦ Working through each part gives: a = {a, b}, B = {x, y}.
Let A={1,2}, B={1,2,3,4}, C={5,6}, D={5,6,7,8}. Verify (i) A x (B intersect C) = (A x B) intersect (A x C). (ii) A x C is a subset of B x D.
Hint. B and C share no elements at all, so their intersection is empty -- work out what that forces on both sides of (i) before computing anything else.
(i) B intersect C is empty (B and C share no elements), so A x (B intersect C) is empty. Separately, A x B has second elements only from {1,2,3,4} and A x C has second elements only from {5,6}, so their intersection is also empty -- both sides match. (ii) A x C = {(1,5),(1,6),(2,5),(2,6)}; every one of these pairs has its first element in B={1,2,3,4} and second element in D={5,6,7,8}, so each pair is automatically in B x D.
✦ Working through each part gives: (i) Both sides equal the empty set, verified. (ii) A x C is indeed a subset of B x D, since B contains A's elements and D contains C's elements.
Let A={1,2} and B={3,4}. Write A x B. How many subsets will A x B have? List them.
Hint. A x B has 4 elements, so it has 2^4 subsets -- list them systematically by how many pairs each contains.
A x B = {(1,3),(1,4),(2,3),(2,4)}, which has 4 elements, so it has 2^4 = 16 subsets, from the empty set up to the full set itself.
✦ Working through each part gives: axB={(1,3),(1,4),(2,3),(2,4)}; 16 subsets: the empty set, the 4 single-pair sets, the 6 two-pair sets, the 4 three-pair sets, and the full set of all 4 pairs.
Let A, B be sets with n(A)=3, n(B)=2. If (x,1), (y,2), (z,1) are in AxB, find A and B, where x, y, z are distinct.
Hint. Every pair in A x B has its first element from A and second element from B -- use the given distinctness and the stated sizes to pin down both sets exactly.
The second elements appearing are 1 and 2, and since n(B)=2, B must be exactly {1,2}. The first elements are x, y, z, stated to be distinct, and since n(A)=3, A must be exactly {x,y,z}.
✦ Working through each part gives: a = {x, y, z}, B = {1, 2}.
A x A has 9 elements, among which are (-1,0) and (0,1). Find A and the remaining elements of A x A.
Hint. 9 = 3x3, so A has exactly 3 elements -- the two given pairs already reveal three of them.
Since n(AxA)=9=3x3, A has exactly 3 elements. The pair (-1,0) shows -1 and 0 are in A; the pair (0,1) shows 1 is in A too. That accounts for all 3 elements, so A={-1,0,1}. Listing all 9 pairs of AxA and removing the two given ones leaves the rest.
✦ Working through each part gives: a = {-1, 0, 1}. The remaining 7 elements of AxA are (-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1).
Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh102.pdf) — three numbered exercises (2.1-2.3, 24 questions) plus the chapter's Miscellaneous Exercise (12 questions); Exercise 2.2 Q4's arrow-diagram figure (Fig 2.7) was rendered directly from the PDF at 250dpi and read visually to confirm the y=x-2 mapping, and the piecewise function boundaries in Miscellaneous Exercise Q1 were re-rendered at high resolution after the raw text extraction garbled the fraction/subscript layout, confirming f's pieces agree at x=3 (function) while g's pieces disagree at x=2 (not a function). Questions are referenced from the NCERT textbook for identification.
Pair self-study with a tutor, a live course or a coaching centre.
