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Cartesian Product of Sets | Algebra - Mathematics PDF Download

We are quite familiar with the term ‘product’ mathematically. It means multiplying. For example, 2 multiplied by 4 gives 8, 16 multiplied by 7 gives 112. Now let’s understand what does the term ‘Cartesian’ stand for? And further, what does a cartesian product mean. Complicated? Let’s find out.

Cartesian Product
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 Sets | Algebra - Mathematics

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 for You
Question: 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 Sets - Algebra - Mathematics

1. What is the Cartesian product of sets in mathematics?
Ans. The Cartesian product of two sets A and B, denoted by A × B, is a set that contains all ordered pairs (a, b) where a belongs to A and b belongs to B. In other words, it is the set of all possible combinations of elements from A and B.
2. How do you compute the Cartesian product of two sets?
Ans. To compute the Cartesian product of two sets A and B, we pair each element of A with each element of B. For example, if A = {1, 2} and 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 the product of their individual cardinalities. In other words, if |A| represents the cardinality of set A and |B| represents the cardinality of set B, then |A × B| = |A| * |B|.
4. Can the Cartesian product of two sets be empty?
Ans. Yes, the Cartesian product of two sets can be empty. This occurs when at least one of the sets in the product is empty. For example, if A = {1, 2} and B = {}, then A × B = {} (empty set).
5. How is the Cartesian product related to combinatorics?
Ans. The Cartesian product is closely related to combinatorics, as it allows us to count the number of possible outcomes in certain situations. By taking the Cartesian product of sets representing different choices or possibilities, we can determine the total number of options available. This concept is fundamental in combinatorial analysis and permutations.
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