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Introduction, Relations | Algebra - Mathematics PDF Download

Relation
Given any two non-empty sets A and B, A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B.
Consider an example of two sets,  A = {2, 5, 7, 8, 9, 10, 13} and B = {1, 2, 3, 4, 5}. The Cartesian product A × B has 30 ordered pairs such as A × B = {(2, 3), (2, 5)…(10, 12)}. From this, we can obtain a subset of A × B, by introducing a relation R between the first element and the second element of the ordered pair (x, y) as
R = {(x, y): x = 4y – 3, x ∈ A and y ∈ B}
Then, R = {(5, 2), (9, 3), (13, 4)}.

Introduction, Relations | Algebra - Mathematics

(Arrow representation of the Relation R) 

Representation of Relation
A relation is represented either by Roster method or by Set-builder method. Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}. The relation is that the elements of A are the square of the elements of B.

  • In set-builder form, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.
  • In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.

Introduction, Relations | Algebra - Mathematics

Terminologies

  • Before getting into details, let us get familiar with a few terms:
  • Image: Suppose we are looking in a mirror. What do we see? An image or reflection. Similarly, for any ordered pairs, in any Cartesian product (say A × B), the second element is called the image of the first element.
  • Domain: The set of all first elements of the ordered pairs in a relation R from a set A to a set B.
  • Range: The set of all second elements in a relation R from a set A to a set B.
  • Codomain: The whole set B. Range ⊆ Codomain.

Total Number of Relations

For two non-empty set, A and B. If the number of elements in A is h i.e., n(A) = h & that of B is k i.e., n(B) = k, then the number of ordered pair in the Cartesian product will be n(A × B) = hk. The total number of relations is 2hk.

Solved Examples for You

Problem: Let A = {5, 6, 7, 8, 9, 10} and B = {7, 8, 9, 10, 11, 13}. Define a relation R from A to B by
R = {(x, y): y = x + 2}. Write down the domain, codomain and range of R.
Solution: Here, R = {(5, 7), (6, 8), (7, 9), (8, 10), (9, 11)}.

Introduction, Relations | Algebra - Mathematics

Domain = {5, 6, 7, 8, 9}
Range = {7, 8, 9, 10, 11}
Co-domain = {7, 8, 9, 10, 11, 13}.

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FAQs on Introduction, Relations - Algebra - Mathematics

1. What is the definition of relations in mathematics?
Ans. In mathematics, relations refer to the connections or associations between two or more elements or sets. It is a way to establish a link or correspondence between different objects or entities.
2. How are relations represented in mathematics?
Ans. Relations can be represented in various ways in mathematics. The most common representation is through ordered pairs or tuples, where each element is related to another element. Additionally, relations can be represented using tables, graphs, or even in verbal or written form.
3. What are the different types of relations in mathematics?
Ans. There are several types of relations in mathematics, including: - Reflexive Relations: Relations where every element is related to itself. - Symmetric Relations: Relations where if element A is related to element B, then element B is also related to element A. - Transitive Relations: Relations where if element A is related to element B, and element B is related to element C, then element A is also related to element C. - Equivalence Relations: Relations that are reflexive, symmetric, and transitive. - Partial Order Relations: Relations that are reflexive, antisymmetric, and transitive.
4. Can you provide an example of a relation in mathematics?
Ans. Yes, consider the relation "less than" or "<" on the set of natural numbers. In this relation, each number is related to another number if it is less than that number. For example, 2 is related to 5 because 2 is less than 5. However, 5 is not related to 2 because 5 is not less than 2.
5. How are relations different from functions in mathematics?
Ans. Relations and functions are related concepts in mathematics, but they have some differences. While both involve a connection between elements, functions are a specific type of relation where each element in one set is related to exactly one element in another set. In other words, functions have a unique output for each input, whereas relations can have multiple outputs for a single input.
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