Sets and Functions:- Enhanced Course with Interactive Elements

**Definition (Natural Numbers - $\mathbb{N}$):**  
The set of counting numbers starting from 0.  
$\mathbb{N} = \{0, 1, 2, 3, 4, ...\}$  

**Definition (Integers - $\mathbb{Z}$):**  
All positive/negative whole numbers and zero.  
$\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$  

**Theorem (Rational Numbers - $\mathbb{Q}$):**  
Numbers of the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$.  
*Example:* $\frac{2}{5}$, $\frac{10}{20} = \frac{1}{2}$ (reduced form).  

**Theorem (Irrational Numbers):**  
Cannot be expressed as $\frac{p}{q}$.  
*Examples:* $\sqrt{2}$, $\pi$.  

**Definition (Real Numbers - $\mathbb{R}$):**  
Union of rational and irrational numbers.  

**Definition (Set):**  
A collection of distinct objects.  
*Notation:* $\{1, 2, 3\}$.  

**Definition (Cardinality):**  
Number of elements in a set.  
*Example:* $|\{1, 2, 3\}| = 3$.  

**Theorem (Subset):**  
$X \subseteq Y$ if every element of $X$ is in $Y$.  
*Example:* $\{1, 2\} \subseteq \{1, 2, 3\}$.  

**Definition (Set Comprehension):**  
Constructs a subset using a rule.  
*Example:* $\{x^2 \mid x \in \mathbb{Z}, x \text{ even}\}$ (squares of even integers).  

**Definition (Cartesian Product):**  
$X \times Y = \{(x, y) \mid x \in X, y \in Y\}$.  
*Example:* $A = \{a, b\}, B = \{1, 2\}$  
$A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$.  

**Definition (Binary Relation):**  
A subset of $X \times Y$.  
*Example:* $R = \{(a, 1), (b, 2)\}$.  

**Definition (Function):**  
A relation where each input maps to exactly one output.  
*Notation:* $f: X \rightarrow Y$.  

**Theorem (Types of Functions):**  
- **Injective:** Each input maps to a unique output.  
- **Surjective:** Co-domain equals the range.  
- **Bijective:** Both injective and surjective.  

**Definition (Composition):**  
$(f \circ g)(x) = f(g(x))$.  
*Example:* $f(x) = 2x$, $g(x) = x + 1$  
$(f \circ g)(x) = 2(x + 1) = 2x + 2$.  

**Definition (Inverse Function):**  
$f^{-1}$ exists if $f$ is bijective.  
*Example:* $f(x) = 3x + 2$, $f^{-1}(y) = \frac{y-2}{3}$.  

**Example (Set Operations):**  
![Venn Diagram](https://byjus.com/maths/wp-content/uploads/2021/11/Venn-Diagram-1.png)  
- **Union (A ∪ B):** All elements in A or B.  
- **Intersection (A ∩ B):** Common elements in A and B.  
- **Complement (A'):** Elements not in A.  

**Question 1:**  
Which set is a subset of $\{2, 3, 4, 5\}$?  
- [ ] $\{2, 6\}$  
- [x] $\{2, 4\}$  
- [ ] $\{5, 7\}$  

**Question 2:**  
What is $\gcd(18, 24)$?  
- [ ] 4  
- [x] 6  
- [ ] 8