Sets and Functions:- Detailed Course with Examples and Practice

1. Numbers and Basic Operations

1.1 Natural Numbers and Integers

• Natural Numbers ($\mathbb{N}$): Counting numbers starting from 0. $\mathbb{N} = {0, 1, 2, 3, 4, …}$
• Integers ($\mathbb{Z}$): All positive, negative whole numbers, and zero. $\mathbb{Z} = {…, -3, -2, -1, 0, 1, 2, 3, …}$

1.1.3 Arithmetic Operations

Practice:

1. Calculate $15 \mod 4$. Answer: $3$ (since $15 ÷ 4 = 3$ remainder $3$)
2. What is $7 × 6$? Answer: $42$

1.1.4 Factors and Multiples

• Factor: $a$ is a factor of $b$ if $b \mod a = 0$.
• Multiple: $b$ is a multiple of $a$.

Examples:

• 2 is a factor of 6 ($6 \mod 2 = 0$).
• 10 is a multiple of 5.

1.2 Rational and Real Numbers

• Rational Numbers ($\mathbb{Q}$): Numbers of the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}, q \neq 0$.
  • Example: $\frac{2}{5}, \frac{10}{20} = \frac{1}{2}$ (after reduction).
• Irrational Numbers: Cannot be written as $\frac{p}{q}$.
  • Examples: $\sqrt{2}, \pi$
• Real Numbers ($\mathbb{R}$): All rational and irrational numbers.

Practice:

1. Is $0.333…$ rational? Answer: Yes, $0.333… = \frac{1}{3}$.

1.2.1 Greatest Common Divisor (gcd)

• Definition: The largest positive integer dividing both $p$ and $q$.
• Examples:
  • $\gcd(9, 12) = 3$
  • $\gcd(15, 45) = 15$
  • $\gcd(0, q) = q$
  • $\gcd(1, q) = 1$

Practice:

1. Find $\gcd(18, 24)$. Answer: $6$

2. Sets

2.1 Definition and Notation

• Set: A collection of well-defined objects.
  • Notation: Curly braces, e.g., ${1, 2, 3}$
  • Elements: Members of a set.

Examples:

• Finite: ${0, 1, 2, …, 9}$ (natural numbers < 10)
• Infinite: ${0, 2, 4, …}$ (even numbers)

2.2 Cardinality

• Cardinality: Number of elements in a set.
  • Example: $|{1, 2, 3}| = 3$

2.3 Subsets

• Subset: $X \subseteq Y$ if every element of $X$ is in $Y$.
• Proper Subset: $X \subset Y$ if $X \subseteq Y$ and $X \neq Y$.

Examples:

• ${1, 2} \subseteq {1, 2, 3}$
• ${1, 2, 3} \subset {1, 2, 3, 4}$

Practice:

1. Is ${2, 4} \subseteq {2, 3, 4, 5}$? Answer: Yes

2.4 Set Comprehension

• Definition: Describes a set by a property.
  • Example: ${x^2 \mid x \in \mathbb{Z}, x \text{ even}}$ (squares of even integers)

3. Relations

3.1 Cartesian Product

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

3.2 Binary Relation

• Definition: A subset of $X \times Y$.
• Example:
  • $A = {a, b}, B = {1, 2, 3}$
  • $R = {(a, 1), (b, 2)}$ is a relation from $A$ to $B$.

3.3 Properties of Relations

Practice:

1. Is the relation $R = {(1,2), (2,1)}$ on ${1,2}$ symmetric? Answer: Yes

4. Functions

4.1 Definition

• Function: A relation where each input has exactly one output.
  • Notation: $f: X \rightarrow Y$
  • Domain: Set $X$ (inputs)
  • Co-domain: Set $Y$ (possible outputs)
  • Range: Actual outputs ${f(x) \mid x \in X}$

Examples:

• $f(x) = x^2$, $x \in \mathbb{R}$, domain: $\mathbb{R}$, range: $[0, \infty)$

4.2 Types of Functions

4.3 Finding Domain and Range

• Example: $f(x) = \sqrt{x}$, domain: $[0, \infty)$
• Example: $f(x) = x^2$, range: $[0, \infty)$, not surjective if co-domain is $\mathbb{R}$

Practice:

1. Find the domain of $f(x) = \frac{1}{x-2}$. Answer: $x \neq 2$

4.4 Function Composition

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

4.5 Inverse Functions

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

5. Practice Questions

Sets and Relations

1. List all subsets of ${a, b}$. Answer: ${}, {a}, {b}, {a, b}$
2. If $A = {1,2}$, $B = {3,4}$, find $A \times B$. Answer: ${(1,3), (1,4), (2,3), (2,4)}$

Functions

3. Is $f(x) = x^2$ injective on $\mathbb{R}$? Answer: No, since $f(2) = f(-2) = 4$.
4. Find the range of $f(x) = |x|$, $x \in \mathbb{R}$. Answer: $[0, \infty)$
5. If $f(x) = 2x + 1$, find $f^{-1}(y)$. Answer: $f^{-1}(y) = \frac{y-1}{2}$

6. Extra Examples

Set Comprehension

• Set of odd numbers less than 10: ${x \mid x \in \mathbb{N}, x < 10, x \mod 2 = 1} = {1, 3, 5, 7, 9}$

Relations

• On $S = {1, 2, 3}$, define $R = {(x, y) \mid x \leq y}$.
  • Is $R$ reflexive? Yes, since $(x, x) \in R$ for all $x$.
  • Is $R$ symmetric? No, since $(1, 2) \in R$ but $(2, 1) \notin R$.

Functions

• If $f(x) = x + 2$, $g(x) = 2x$, find $(g \circ f)(x)$.
  • $g(f(x)) = 2(x + 2) = 2x + 4$

7. Answers to Practice Questions

1. $15 \mod 4 = 3$
2. $7 \times 6 = 42$
3. $\gcd(18, 24) = 6$
4. Subsets of ${a, b}$: ${}, {a}, {b}, {a, b}$
5. $A \times B = {(1,3), (1,4), (2,3), (2,4)}$
6. $f(x) = x^2$ is not injective on $\mathbb{R}$
7. Range of $f(x) = |x|$ is $[0, \infty)$
8. $f^{-1}(y) = \frac{y-1}{2}$

8. Layout/UI Suggestions

• Use color-coded boxes for definitions, theorems, and examples.
• Add collapsible sections for practice questions and answers.
• Include diagrams for set relations, function mappings, and Venn diagrams.
• Provide interactive quizzes after each section for self-assessment.

This structure covers the essentials of sets and functions, with added clarity, examples, and practice. For a more interactive experience, consider using digital flashcards or quizzes after each topic.