Cartesian Product is one of the operations performed on sets. Set is a collection of well-defined objects. Cartesian product is nothing but multiplying two or more sets to get the product set. It is also known as the cross product. Get to know about the Cartesian Product of Two Sets definition, solved examples, and what is an ordered pair and others in the following sections.
Cartesian Product of Two Sets – Definition
The cartesian product of two sets A and B is denoted by A x B. Where A x B is a set of all possible ordered pairs in the form of (a, b), here a ∈ A, b ∈ B. The roster form of the cartesian product of two sets is A x B = {(a, b) | a ∈ A and b ∈ B}. The cartesian product is also called the cross product.
The cartesian product of two sets A x B is not equal to B x A. Because their ordered pairs are not equal. If A = B, then the cartesian product A x B = A x A = A² is called the cartesian square. A² = {(a, b) : a ∈ A and b ∈ A}. Follow these sections to learn the concept of the ordered pair in sets.
Ordered Pairs
Ordered pairs are formed when you perform cross product between two sets. Ordered pairs are the pairs of numbers with coordinates to represent various points on the coordinate place. It is defined as the set of two objects gathered with an order associated with them. These are usually, written in parenthesis and each element are separated by a comma. In an ordered pair (a, b) the element a is called the first component or first entry and element b is called the second component or second entry of the pair.
Two ordered pairs are said to be equal if their corresponding entries are equal. So, (a, b) ≠ (b, a). The equality of ordered pairs is (a, b) = (d, c) only when a = d and b = c.
If an ordered pair has more than two elements in it, then it is called an ordered n-tuple. An ordered n-tuple is formed when a set of n objects are grouped with an order associated with them. Tuples are denoted by (a1, a2, a3, a4, . . . an). Ordered pairs are also called 2-tuples.
Properties of Cross Product
- The cartesian product is non-commutative.
- A x B ≠ B x A.
- A x B = B x A only when A = B.
- A x B = ∅, if either A = ∅ or B = ∅
- The cartesian product is non-associative:
- (A x B) x C ≠ A x (B x C)
- Distributive Property over Set Union is
- A x (B U C) = (A x B) U (A x C)
- Distributive Property over Set Intersection is
- A x (B ∩ C) = (A x B) ∩ (A x C)
- Distributive Property over Set Difference is
- A x (B – C) = (A x B) – (A x C)
- If A ⊆ B, then A × C ⊆ B × C for any set C.
Cartesian Product of Several Sets
Cartesian product of several sets means the product of more than two sets. The cross product of n non-empty sets is A₁ x A₂ x A₃ x . . . x An is defined as the set of ordered n-tuples (a₁, a₂, a₃, . . an) where ai ∈ Ai. If A₁ = A₂ = A₃ . . . = An = A then A₁ x A₂ x A₃ x . . . x An is the nth cartesian power of set A and is denoted as An.
Also, Read:
| Intersection of Sets | Operations on Sets |
| Union of Sets | Complement of a Set |
| Difference of Two Sets |
Cartesian Product of Two Sets Examples
Question 1:
If A = {4, 8, 15}, B = {x, y, z}, then find A x B and B x A.
Solution:
Given two finite non-empty sets are
A = {4, 8, 15}, B = {x, y, z}
A x B = {4, 8, 15} x {x, y, z}
= {(4, x), (4, y), (4, z), (8, x), (8, y), (8, z), (15, x), (15, y), (15, z)}
B x A = {x, y, z} x {4, 8, 15}
= {(x, 4), (x, 8), (x, 15), (y, 4), (y, 8), (y, 15), (z, 4), (z, 8), (z, 15)}
The 9 ordered pairs thus formed can represent the position of points in a plane, if A and B are subsets of a set of real numbers.
Question 2:
If A = {a, b, c, d}, then find A x A or A².
Solution:
Given set is A = {a, b, c, d}
A² = A x A = {a, b, c, d} x {a, b, c, d}
= {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b), (b, c), (b, d), (c, a), (c, b), (c, c), (c, d), (d, a), (d, b), (d, c), (d, d)}
The number of ordered pairsare 16.
Question 3:
If A and B are two sets and A × B consists of 6 elements: If three elements of A × B are (2, 5) (3, 7) (4, 7) find A × B.
Solution:
Given that,
A × B consists of 6 elements.
Three elements of A × B are (2, 5) (3, 7) (4, 7)
By observing the above-ordered pairs, we can say that 2, 3, 4 are the elements of setA and 5, 7 are the elements of set B.
So, A = {2, 3, 4}, B = {5, 7}
Then A x B = {(2, 5), (2, 7), (3, 5), (3, 7), (4, 5), (4, 7)}
Thus A x B has 6 elements.
Question 4:
If A = {2, 4, 6}, B = {1, 3, 5}, then find (i) A x B (ii) B x B (iii) B x A (iv) A x A.
Solution:
Given two sets are A = {2, 4, 6}, B = {1, 3, 5}
(i) A x B = {2, 4, 6} x {1, 3, 5}
= {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)}
(ii) B x B = {1, 3, 5} x {1, 3, 5}
= {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}
(iii) B x A = {1, 3, 5} x {2, 4, 6}
= {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)}
(iv) A x A = {2, 4, 6} x {2, 4, 6}
= {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}.
FAQs on Cartesian Product of Two Sets
1. What is the Cartesian product of 3 sets?
The cartesian product of three sets A, B and C are denoted by A x B x C = {(a, b, c) | a ∈ A, b ∈ B, c ∈ C}.
2. What is the cartesian square?
Cartesian square means the cross product of two same sets. A² is the cartesian square.
3. What is the product of two sets?
The product is nothing but the cross product of two sets A and B, denoted A × B. It is the set of all possible ordered pairs where the elements of A are first and the elements of B are second.
4. Write an example for the cartesian product?
Let A = {1, 2, 3}, B = {a, b, c}, C = {apple, grapes, guava} be three sets.
A x B x C = {(1, a, apple), (1, a, grapes), (1, a, guava), (1, b, apple), (1, b, grapes), (1, b, guava), (1, c, apple), (1, c, grapes), , (1, c, guava), (2, a, apple), (2, a, grapes), (2, a, guava), (2, b, apple), (2, b, grapes), (2, b, guava), (2, c, apple), (2, c, grapes), (2, c, guava), (3, a, apple), (3, a, grapes), (3, a, guava), (3, b, apple), (3, b, grapes), (3, b, guava), (3, c, apple), (3, c, grapes), (3, c, guava)}
It has 27 ordered pairs.