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Cartesian Product of Sets, Ordered Pairs, Properties of Cartesian product, Practice Problems & FAQs

Cartesian Product of Sets, Ordered Pairs, Properties of Cartesian product, Practice Problems & FAQs

We are already familiar with the product of numbers. But do you know that we can find the product of sets too and just like in the case of numbers, we can find the product of  many sets at a time.

In set theory, the cartesian product of two or more sets is the product of all sets in an ordered way. To understand cartesian products, first, we will learn the concept of ordered pairs.

Table of Contents

Ordered Pairs

An ordered pair consists of two objects or elements in a fixed order.

Let be an ordered pair. In an ordered pair, the positions of the elements are fixed, here is the first element and is the second element of the ordered pair Also, the position of every possible point in a two-dimensional plane is represented by an ordered pair.

Example: The points  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo>&#xA0;</mo><mi>B</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo>&#xA0;</mo><mi>C</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo>&#xA0;</mo><mi>D</mi><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mi>E</mi><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math>in the cartesian plane are represented in the form of ordered pairs.

Note

  • and are two different ordered pairs
  • is known as an ordered pair. Similarly is known as an ordered triplet.

Equality of Ordered pairs

Two ordered pairs are equal if and only if their corresponding first and second elements are equal.


<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>&#xA0;</mo><mo>,</mo><msub><mi>b</mi><mn>1</mn></msub><mo>)</mo><mo>=</mo><mo>(</mo><msub><mi>a</mi><mn>2</mn></msub><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><msub><mi>b</mi><mn>2</mn></msub><mo>)</mo><mo>&#xA0;</mo><mo>&#x21D4;</mo><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><msub><mi>a</mi><mn>2</mn></msub><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><msub><mi>b</mi><mn>1</mn></msub><mo>=</mo><msub><mi>b</mi><mn>2</mn></msub></math>

Example: Consider two points & whose position is given by the coordinates , respectively. As y-coordinate of point of y-coordinate of point  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x21D2;</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>&#x2260;</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>Therefore, these ordered pairs are not equal.

Cartesian Product of Sets

The cartesian product of two sets results in a set which is the collection of all ordered pairs.

Let’s understand this with the help of an example.

Let be two sets such that,  & 

Let’s find the number of ordered pairs that we can make from these two sets respectively. Proceeding in a quite thorough manner, we can recognize that there will be six different pairs. They can be written as given below:

The above-ordered pairs represent the Cartesian product of given two sets .

Hence, If and be two non-empty sets then the set of all ordered pairs where and is called the cartesian product of sets & and is denoted by .

Similarly, 

Example: If Set , Set , then 

                   

Note: 

  • The Cartesian product is also called the cross product, set direct product or the product set of and .
  • The Cartesian product can be found for more than two sets also.

Number of Elements in the Cartesian Product of Sets

If and are two finite sets such that and

  

Example: and

and

Number of elements in  i.e.

Number of elements in  i.e.

Cartesian Product of Empty Set

As we know, an empty set does not have any element in it. The cardinality or the size of an empty set is also zero. The cartesian product of a set, say and the empty set, is an empty set only.

.

Note: If one of the sets is infinite, then is an infinite set.

Cartesian Square

If both the sets of a cartesian product are the same, say , then the cartesian product of set is called a cartesian square.

Properties of Cartesian Product

  • Cartesian product is not commutative. Thus, if we change the order of sets the result changes. If are two sets, then the cartesian product of
  • Cartesian product is not associative. If we regroup the sets in the cartesian product, then it will change the result. If are three sets, then
  • Distribution property of cartesian product over the intersection of sets is given by 

  • Distribution property of cartesian product over the union of sets is given by

 

Cartesian Product of Three sets

The Cartesian product of three sets can be explained using an example.

Consider three sets

Now, we need to find the cartesian product of these three sets.

The number of ordered triplets in {since the number of elements in each of the given three sets is 2}

Thus, the ordered triplets of can be tabulated as:

Elements Elements to be selected from sets Ordered pairs
1st element {1, 2} × {3, 4} × {5, 6} (1, 3, 5)
2nd element {1, 2} × {3, 4} × {5, 6} (1, 3, 6)
3rd element {1, 2} × {3, 4} × {5, 6} (1, 4, 5)
4th element {1, 2} × {3, 4} × {5, 6} (1, 4, 6)
5th element {1, 2} × {3, 4} × {5, 6} (2, 3, 5)
6th element {1, 2} × {3, 4} × {5, 6} (2, 3, 6)
7th element {1, 2} × {3, 4} × {5, 6} (2, 4, 5)
8th element {1, 2} × {3, 4} × {5, 6} (2, 4, 6)

Therefore,

Practice Problems

Q 1. Let be a non empty set such that has 9 elements among which two elements are found to be and then set is

                            

                       

Answer: We have, = 9

Now,

 

 

Also and

Hence,

Q 2. Let be the set of all divisors of 8 and be the set of all the divisors of 10, then find the number of elements in .

Answer: Let is the set of all divisors of 8

Then,

Let is the set of all divisors of 10

Then,

Therefore, the number of elements in is 16.

Q 3. If , find and

Answer: Two ordered pairs are equal when their corresponding elements are equal.

On equating the elements of we get,

FAQs

Q 1. Does the ordering of sets matter in the Cartesian product?

Answer: Yes, as the Cartesian product is not commutative so it depends on the ordering of the sets

Q 2. Can we find the cartesian product of two empty sets?

Answer: Cartesian product of two empty sets is possible. The obtained cartesian product would also be empty.

Q 3. What is the application of the Cartesian product?

Answer: The images that we see on our mobile phones are in the form of pixels which are nothing but the graphical representation of Cartesian products.

Q 4. What is the difference between relations and cartesian product?

Answer: A relation is a subset of the Cartesian product.

Related Concept Links
Set Theory Relations
Functions: Domain, Codomain, and Range One-One & Onto and Functions
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