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Cartesian Product of Sets
Before getting familiar with this term, let us understand what does Cartesian mean. Remember the terms used when plotting a graph paper like axes (x-axis, y-axis), origin etc. For example, (2, 3) depicts that the value on the x-plane (axis) is 2 and that for y is 3 which is not the same as (3, 2).
The way of representation is fixed that the value of the x coordinate will come first and then that for y (ordered way). Cartesian product means the product of the elements say x and y in an ordered way.

Cartesian Product of Two Sets | Mathematics (Maths) Class 12 - JEE

Cartesian Product of Sets
The Cartesian products of sets mean the product of two non-empty sets in an ordered way. Or, in other words, the collection of all ordered pairs obtained by the product of two non-empty sets. An ordered pair means that two elements are taken from each set.
For two non-empty sets (say A & B), the first element of the pair is from one set A and the second element is taken from the second set B. The collection of all such pairs gives us a Cartesian product.
The Cartesian product of two non-empty sets A and B is denoted by A × B. Also, known as the cross-product or the product set of A and B. The ordered pairs (a, b) is such that a ∈ A and b ∈ B. So, A × B = {(a,b): a ∈ A, b ∈ B}. For example, Consider two non-empty sets A = {a1, a2, a3} and B = {b1, b2, b3}

Cartesian product A×B = {(a1, b1), (a1, b2), (a1, b3), ( a2, b1), (a2, b2),(a2, b3), (a3, b1), (a3, b2), (a3, b3)}.
It is interesting to know that (a1, b1) will be different from (b1, a1). If either of the two sets is a null set, i.e., either A = Φ or B = Φ, then, A × B = Φ i.e., A × B will also be a null set

Number of Ordered Pairs
For two non-empty sets, A and B. If the number of elements of A is h i.e., n(A) = h & that of B is k i.e., n(B) = k, then the number of ordered pairs in Cartesian product will be n(A × B) = n(A) × n(B) = hk.

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 = Set of first elements = {m, n, x, y} and Q = Set of second elements = {1, -1}

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FAQs on Cartesian Product of Two Sets - Mathematics (Maths) Class 12 - JEE

1. What is the Cartesian product of two sets?
Ans. The Cartesian product of two sets, A and B, denoted as A × B, is a mathematical operation that combines every element of set A with every element of set B to create a new set. In other words, it is the set of all possible ordered pairs (a, b), where a is an element of A and b is an element of B.
2. How do you calculate the Cartesian product of two sets?
Ans. To calculate the Cartesian product of two sets, list all the elements of the first set and pair them with every element of the second set. For example, if set A = {1, 2} and set B = {a, b}, the Cartesian product A × B would be {(1, a), (1, b), (2, a), (2, b)}.
3. What is the cardinality of the Cartesian product of two sets?
Ans. The cardinality of the Cartesian product of two sets A and B is equal to the product of the cardinalities of A and B. In other words, if set A has n elements and set B has m elements, then the cardinality of A × B is n * m.
4. Can the Cartesian product of two sets be empty?
Ans. Yes, the Cartesian product of two sets can be empty if either one or both of the sets are empty. For example, if set A is empty and set B = {1, 2}, then the Cartesian product A × B would be an empty set.
5. How is the Cartesian product related to combinatorics?
Ans. The Cartesian product is closely related to combinatorics, which is the branch of mathematics that deals with counting, arranging, and choosing elements from sets. Combinatorics often involves calculating the number of possible outcomes in various scenarios, and the Cartesian product provides a way to combine elements from different sets to create all possible outcomes.
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