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Relations and Functions

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The "Relations" deals with understanding the connections or relationships between different sets of elements. A relation is a way to link elements of one set to elements of another set.

To understand relations, it is essential to have a good understanding of sets. A set is a collection of distinct objects. For example, the set of all vowels in English language can be represented as: {a, e, i, o, u}.

Now, suppose we have two sets A and B. A relation R between A and B is a subset of the Cartesian product A × B, which is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. In simpler terms, a relation R is a way to connect each element of set A with one or more elements of set B.

Let's take an example:

R = {(1,2), (2,4), (3,6)}

In this Example,

Consider two sets A = {1, 2, 3} and B = {2, 4, 6}. The relation R between A and B such that each element of A is related to its double in B can be represented as: R = {(1,2), (2,4), (3,6)}

Here, we can see that each element of set A is related to its double in set B. So, 1 is related to 2, 2 is related to 4 and 3 is related to 6.

There are different types of relations that can exist between two sets. Some of the commonly studied relations are:

  1. Reflexive Relation: A relation R on a set A is reflexive if every element of A is related to itself. For example, the relation R = {(1,1), (2,2), (3,3)} is reflexive on the set A = {1, 2, 3}.

  2. Symmetric Relation: A relation R on a set A is symmetric if for every (a, b) ∈ R, (b, a) ∈ R. In simpler terms, if a is related to b, then b is also related to a. For example, the relation R = {(1,2), (2,1), (2,3), (3,2)} is symmetric on the set A = {1, 2, 3}.

  3. Transitive Relation: A relation R on a set A is transitive if for every (a, b) ∈ R and (b, c) ∈ R, (a, c) ∈ R. In simpler terms, if a is related to b and b is related to c, then a is also related to c. For example, the relation R = {(1,2), (2,3), (1,3)} is transitive on the set A = {1, 2, 3}.

Functions

A function is a special type of relation where each element in the domain is paired with exactly one element in the range. In other words, a function maps each element in the domain to a unique element in the range.

For example, the set of ordered pairs {(1, 2), (2, 4), (3, 6)} is a function because each element in the domain is mapped to a unique element in the range.

Functions can be represented in different ways, such as through a graph or an equation. For example, the function f(x) = 2x maps each value of x to its double in the range. If we plot the points (1, 2), (2, 4), and (3, 6) on a graph, we can see that they form a straight line passing through the origin.

We can also perform operations on functions, such as adding, subtracting, multiplying, and dividing them. For example, if we have two functions f(x) = 2x and g(x) = x + 1, we can add them together to get h(x) = f(x) + g(x) = 3x + 1.

It is important to note that some relations are not functions, such as {(1, 2), (2, 4), (3, 2)} where the element 2 in the range is paired with two different elements in the domain. Also, some functions have restrictions on their domain or range, which we must consider when working with them.

Function:

f(x) = x^2


 

For example,

This is a function because each value of x is mapped to a unique value of y. For example, if we substitute x = 2 into the equation, we get f(2) = 2^2 = 4. If we substitute x = 3, we get f(3) = 3^2 = 9.

It is also possible to represent a function as a set of ordered pairs, for example:

Function: {(1, 3), (2, 4), (3, 5)}

This is a function because each element in the domain is mapped to a unique element in the range. For example, the element 2 in the domain is mapped to the element 4 in the range. We can also write this function as f(x) = x + 2, where each value of x is added to 2 to get the corresponding value of y.