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Cartesian Product of Sets

Last Updated : 16 May, 2024
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Product‘ mathematically signifies the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5. One must be familiar with the basic operations on sets like Union and Intersection, which are performed on 2 or more sets. Cartesian Product is also one such operation that is performed on two sets, which returns a set of ordered pairs.

In this article, we have covered, ordered pair definition, cartesian product of sets, and others in detail.

What is an Ordered Pair?

An ordere­d pair has two parts. The first part is called the first compone­nt. The second part is called the­ second component. We write­ an ordered pair like this: (a, b). The­ letter ‘a’ is the first compone­nt. The letter ‘b’ is the­ second component. An ordere­d pair has two things. One thing comes first. The othe­r thing comes second.

Example:

(5, 7) is an ordered pair of integers.

Note: (5, 7) ≠ (7, 5), an ordered pair (a, b) is equal to (x, y) only if a = x and b = y.

Cartesian Product of Sets

When two se­ts have items in them, A and B, the­ir Cartesian product is all the pairs you can make. One­ part of the pair comes from set A. The­ other part comes from set B. We­ make every possible­ pair this way. The result is a new se­t of all these pairs. We write­ this new set as A×B.

A × B = {(a, b) : a ∈ A and b ∈ B}

Example: 

Let A = {1, 2} and B = {4, 5, 6}

A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)}

Here the first component of every ordered pair is from set A the second component is from set B.

Cartesian Product of two sets can be easily represented in the form of a matrix where both sets are on either axis, as shown in the image below. Cartesian Product of  A = {1,  2} and B = {x, y, z}

Cartesian Product of two sets

Properties of Cartesian Product

Various properties of cartesian product includes,

1. Cartesian Product is non-commutative: A × B ≠ B × A

Example: 

A = {1, 2} , B = {a, b}

A × B = {(1, a), (1, b), (2, a), (2, b)}

B × A = {(a, 1), (b, 1), (b, 1), (b, 2)}

Therefore as A ≠ B we have A × B ≠ B × A

2. A × B = B × A, only if A = B

Proof:

Let A × B = B × A then we have  

A ⊆  B  and B ⊆  A, it follows that A = B

3. Cardinality of Cartesian Product is defined as number of elements in A × B and is equal to the product of cardinality of both sets i.e.,

|A × B| = |A| × |B|

Proof:

Let a ∈ A then the number of ordered pair (a, b) such that b ∈ B is |B|

Therefore we have |B| choices for b for each a where a ∈ A therefore the number of element in A × B is |A| × |B|

4. A × B = ∅, if either A = ∅ or B = ∅

Proof: 

Suppose A×B=∅. This means there are no ordered pairs (a,b) where a∈A and b∈B.

If A is non-empty, then there exists at least one element a∈A. For any such a, there should be an ordered pair (a,b) for some b∈B, as B is not empty. But since we have assumed A×B=∅, this is a contradiction. Hence, A must be empty.

Similarly, if B is non-empty, then there exists at least one element b∈B. For any such b, there should be an ordered pair (a,b) for some a∈A, as A is not empty. But since we have assumed A×B=∅, this is a contradiction. Hence, B must be empty.

Therefore, if 𝐴 × 𝐵 = ∅, either A or B must be empty

Hence, the statement 𝐴 × 𝐵 = ∅ if and only if either A=∅ or 𝐵 = ∅ is proven.

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Problems on Cartesian Product of Sets

Problem 1: Find the value of x and y given (2x – y,  25) = (15,  2x + y)?

Solution:  

As we know from the property of ordered pairs, 2x – y = 15 and 25 = 2x + y.

Solving the linear equations we have x = 10 and y = 5.

Problem 2. Given A = {2, 3, 4 , 5} and B = {4 , 16 , 23}, a ∈ A, b ∈ B, find the set of ordered pairs such that a2 < b?

Solution:

As 22 < 16 and 23, 32 < 16 and 23, 42 < 23  

We have the set of ordered pairs such that a2 < b is {(2, 16), (2, 23), (3, 16), (2, 23), (4, 23)}

Problem 3. If A = {9, 10} and B = {3, 4, 6}, find A × B and |A × B|? 

Solution:

A × B = {(9, 3), (9, 4), (9, 6), (10, 3), (10, 4), (10, 6)}

|A × B| = |A| * |B| = 2 * 3 = 6

Problem 4. If A × B = {(a, x), (a, y ), (b, x ), (b, y)}, find A and B?

Solution:

We know A is the set of all first components in ordered pairs of A × B and 

B is the set of the second component in the ordered pair of A × B.

Therefore A = {a, b} and B = {x, y}

Problem 5. Given A × B has 15 ordered pairs and A has 5 elements, find the number of elements in B?

Solution:

We know |A × B| = |A| * |B|, 15 = 5 * |B|

Therefore B has 15 / 5 = 3 elements.

Practice Problems on Cartesian Product of Sets

Problem 1: Let A = {a, b, c} and B = {1, 2}. Find A × B.

Problem 2: Let A = {} and B = {x, y}. Find A × B.

Problem 3: Let A = {apple, banana} and B = {red, yellow}. Find A × B.

Problem 4: Let A = {1, 2, 3} and B = {3, 4, 5}. Find A × B.

Problem 5: Let A = {cat} and B = {dog}. Find A × B.

FAQs on Cartesian Product of Sets

What is cartesian product of two sets i.e., A×B?

Cartesian product of sets A and B, denoted A×B, is the set of all possible ordered pairs where the first element is from A and the second from B.

Define ordered pair.

An ordered pair is a pair of elements (a, b) in which the order of the elements is significant. This means (a, b) is distinct from (b, a) if a is not equal to b.

What is cartesian product of 3 sets?

Cartesian product of three sets A, B, and C is the set of all possible ordered triples where first element is from A, second from B, and third from C.

Write formula for cartesian product of sets.

Cartesian product of two sets A and B is defined as:

A × B = {(a, b) | a ∈ A and b ∈ B}

For three sets A, B, and C:

A × B × C = {(a, b, c) | a ∈ A, b ∈ B, and c ∈ C}

What is cartesian product of a set and a null set?

Cartesian product of a set A and a null set (∅) is always an empty set (∅), as there are no elements in the null set to form pairs with elements from set A.



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