Class 12 relation and function

Steps to Solve

1. Define a Relation

A relation in mathematics is a set of ordered pairs. If we have two sets, say set A and set B, a relation from A to B is a subset of the Cartesian product $A \times B$. For example, if $A = {1, 2}$ and $B = {3, 4}$, then a possible relation could be $R = {(1, 3), (2, 4)}$.

2. Define a Function

A function is a special type of relation where each element in set A is related to exactly one element in set B. This means that for every input (from set A), there is one and only one output (in set B). You can represent a function as $f: A \rightarrow B$. For example, if $f(1) = 3$ and $f(2) = 4$, it illustrates a function.

3. Key Difference between Relations and Functions

The key difference is that a function is a type of relation with a specific requirement: it cannot have the same first element in multiple ordered pairs. If a relation includes $(1, 3)$ and $(1, 4)$, it is not a function because the first element (1) is associated with two different second elements.

4. Examples

Consider the following examples to illustrate the concept:

• Relation: $R = {(1, 2), (2, 3), (1, 4)}$ is a relation but not a function because 1 is related to both 2 and 4.
• Function: $f = {(1, 2), (2, 3)}$ is a function because each input has one unique output.

5. Visual Representation

You can visualize relations and functions using graphs. In a graph, if any vertical line intersects a curve more than once, the curve does not represent a function. This is known as the vertical line test.