Cartesian Product of Two Sets - (Maths) Class 12 - JEE PDF Download

Cartesian Product of Sets

Cartesian refers to the system of coordinates used in the plane where each point is represented by an ordered pair (x, y). The order of the two entries matters: (2, 3) is not the same as (3, 2). The Cartesian product of two sets captures this idea by forming ordered pairs whose first element comes from the first set and whose second element comes from the second set.

Definition

Let A and B be two non-empty sets. The Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs whose first component belongs to A and whose second component belongs to B.

Symbolically:

A × B = {(a, b) : a ∈ A and b ∈ B}.

Examples of Cartesian Product

• Let A = {1, 2} and B = {x, y, z}. Then A × B = {(1, x), (1, y), (1, z), (2, x), (2, y), (2, z)}.
• Let A = {a1, a2, a3} and B = {b1, b2, b3}. Then A × B = {(a1, b1), (a1, b2), (a1, b3), (a2, b1), (a2, b2), (a2, b3), (a3, b1), (a3, b2), (a3, b3)}.
• If A = {1, 2} and B = {1, 2}, then A × B contains (1,1), (1,2), (2,1), (2,2). Note that (1, 2) is different from (2, 1).

Key Properties

• Order matters: (a, b) and (b, a) are generally different unless a = b.
• Cardinality (finite sets): If n(A) = h and n(B) = k, then n(A × B) = n(A) × n(B) = hk.
• Null set: If A = ∅ or B = ∅, then A × B = ∅.
• Non-commutativity: In general, A × B ≠ B × A. However, there is a natural correspondence between the two by swapping coordinates.
• Cartesian product with itself: A × A is often written as A2 and consists of ordered pairs both of whose entries belong to A.
• Generalisation to n-tuples: A1 × A2 × ... × An is the set of ordered n-tuples (a1, a2, ..., an) with ai ∈ Ai.

Counting Ordered Pairs - Reasoning

Suppose A has h elements and B has k elements. For each fixed element a ∈ A, we can pair it with every one of the k elements of B. Thus for that fixed a there are k ordered pairs of the form (a, b) with b ∈ B. Since there are h choices for a, the total number of ordered pairs is h × k. Hence n(A × B) = n(A) × n(B).

Connection with Coordinate Geometry

The Cartesian product R × R is the set of all ordered pairs of real numbers and corresponds to the set of points in the Euclidean plane. A point with coordinates (x, y) is an element of R × R. More generally, the Cartesian product gives a formal set-theoretic foundation for coordinates and points in geometry.

Cartesian Product and Relations / Functions

A relation from A to B is any subset of A × B. A function from A to B is a special relation in which each element of A appears as the first component of exactly one ordered pair of the subset. Thus Cartesian products provide the universe in which relations and functions are defined.

Finite and Infinite Sets

• If A and B are finite, n(A × B) = n(A)n(B).
• If either A or B is infinite, then A × B is infinite. For example, R × R is an uncountable set corresponding to all points in the plane.

More Examples

• Let A = {p, q} and B = {0}. Then A × B = {(p, 0), (q, 0)} and n(A × B) = 2 × 1 = 2.
• Let A = ∅ and B = {1, 2}. Then A × B = ∅.

Solved Example

Q.1. Let P & Q be two sets such that n(P) = 4 and n(Q) = 2. If in the Cartesian product we have (m, 1), (n, -1), (x, 1), (y, -1). Find P and Q, where m, n, x, and y are all distinct.

Solution. 

P is the set of the first components of the given ordered pairs.

P = {m, n, x, y}.

Q is the set of the second components of the given ordered pairs.

Q = {1, -1}.

Remarks and Common Mistakes

• Do not confuse A × B with the set product of elements (there is no multiplication of elements unless the elements themselves are numbers and such multiplication is separately defined).
• Remember that the ordered pair (a, b) is not the same object as the unordered set {a, b} unless a = b and an identification is explicitly given.
• Be careful when asked to list elements of A × B - list all combinations of first elements from A with second elements from B.

Summary

The Cartesian product A × B is the collection of all ordered pairs (a, b) with a ∈ A and b ∈ B. For finite sets, its cardinality equals the product of the cardinalities of the factors. Cartesian products form the basic framework for coordinates, relations and functions in mathematics.