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1800-102-2727We 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
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 in the cartesian plane are represented in the form of ordered pairs.
Note
Two ordered pairs are equal if and only if their corresponding first and second elements are equal.
Example: Consider two points & whose position is given by the coordinates , respectively. As y-coordinate of point of y-coordinate of point Therefore, these ordered pairs are not equal.
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:
If and are two finite sets such that and
Example: and
and
Number of elements in i.e.
Number of elements in i.e.
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.
If both the sets of a cartesian product are the same, say , then the cartesian product of set is called a cartesian square.
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,
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,
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 |