Table of Contents
1. Sets
Before studying relations and functions, we must understand sets.
A set is a well-defined collection of distinct objects.
Example:
If \( A = \{1, 2, 3\} \), then the elements of set \(A\) are 1, 2, and 3.
2. Cartesian Product of Sets
If \(A\) and \(B\) are two non-empty sets, then the Cartesian product of
\(A\) and \(B\) is denoted by \(A \times B\) and is defined as:
\[
A \times B = \{(a, b) \mid a \in A, b \in B\}
\]
Solved Example 1
Let \( A = \{1, 2\} \) and \( B = \{3, 4\} \).
Then,
\[
A \times B = \{(1,3), (1,4), (2,3), (2,4)\}
\]
3. Relations
A relation from a set \(A\) to a set \(B\) is a subset of
\(A \times B\).
If \((a, b)\) belongs to a relation \(R\), then we say that
\(a\) is related to \(b\).
Solved Example 2
Let \( A = \{1, 2, 3\} \).
Define a relation \(R\) such that:
\[
R = \{(x, y) \mid y = x^2\}
\]
Then,
\[
R = \{(1,1), (2,4), (3,9)\}
\]
4. Domain, Codomain, and Range
- Domain: Set of all first elements of ordered pairs.
- Codomain: The set to which elements are related.
- Range: Set of all second elements of ordered pairs.
Solved Example 3
For the relation:
\[
R = \{(1,2), (2,4), (3,6)\}
\]
- Domain = \(\{1,2,3\}\)
- Codomain = \(\{2,4,6\}\)
- Range = \(\{2,4,6\}\)
5. Functions
A function is a special type of relation in which each element of the domain is related
to exactly one element of the codomain.
Mathematically, a function \(f\) from set \(A\) to set \(B\) is written as:
\[
f : A \rightarrow B
\]
Solved Example 4
Let \( f(x) = 2x + 1 \) and \( x \in \{1, 2, 3\} \).
Then,
- \(f(1) = 3\)
- \(f(2) = 5\)
- \(f(3) = 7\)
Hence, \(f = \{(1,3), (2,5), (3,7)\}\) is a function.
6. Types of Functions
(a) One-One (Injective) Function
A function is one-one if different elements of the domain have different images.
(b) Onto (Surjective) Function
A function is onto if every element of the codomain has at least one pre-image.
(c) Into Function
A function is into if at least one element of the codomain has no pre-image.
7. Real-Valued Functions
A function whose domain and codomain are subsets of real numbers is called a
real-valued function.
Solved Example 5
Let \( f(x) = x^2 \) where \( x \in \mathbb{R} \).
The range of \(f(x)\) is:
\[
[0, \infty)
\]