M1 VOL1 SETS & FUNCTIONS.pdf

explain all concepts from this pdf with detailed structure , examples, related questions & answers with ease to understand

Here’s a detailed, structured explanation of all the key concepts from the provided PDF, “Mathematics for Data Science-1: Sets and Functions,” along with examples, related questions, and answers, designed for easy understanding¹.

---

1. Set Theory

Definition: A set is a collection of well-defined, distinct objects called elements. Order and duplicates do not matter.

Examples:

• Finite set: Months of the year = {January, February, …, December}
• Infinite set: Even natural numbers = {0, 2, 4, 6, …}

Subsets: A set $X$ is a subset of $Y$ ($X \subseteq Y$) if every element of $X$ is in $Y$. If $X \neq Y$, it is a proper subset ($X \subset Y$).

Example: $X = {1,2,5,7,9,300}, Y = {0,1,2,5,7,9,40,170,300}$ Here, $X \subset Y$.

Set Comprehension: Constructing a subset using a rule, e.g., squares of even integers: ${x^2 \mid x \in \mathbb{Z}, x \text{ is even}}$.

Related Question: What is the cardinality of the set $A = {a, b, c, d}$? Answer: 4

---

2. Natural Numbers, Integers, Rational and Real Numbers

Natural Numbers ($\mathbb{N}$): ${0, 1, 2, 3, …}$

Integers ($\mathbb{Z}$): ${…, -2, -1, 0, 1, 2, …}$

Rational Numbers ($\mathbb{Q}$): Numbers expressible as $\frac{p}{q}$, where $p, q \in \mathbb{Z}, q \neq 0$.

Real Numbers ($\mathbb{R}$): All rational and irrational numbers (e.g., $\sqrt{2}, \pi$).

Example: Is $\frac{2}{4}$ a rational number? Answer: Yes, since it equals $\frac{1}{2}$.

Related Question: What is the greatest common divisor (gcd) of 9 and 12? Answer: 3

---

3. Relations and Functions

Cartesian Product: Given sets $X$ and $Y$, $X \times Y = {(x,y) \mid x \in X, y \in Y}$.

Example: $A = {a, b}, B = {1,2,3}$ $A \times B = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}$

Binary Relation: A subset of $X \times Y$. Example: $R_1 = {(a,1), (b,2)}$ is a relation from $A$ to $B$.

Properties of Relations:

• Reflexive: Every element is related to itself.
• Symmetric: If $(x,y) \in R$, then $(y,x) \in R$.
• Transitive: If $(x,y) \in R$ and $(y,z) \in R$, then $(x,z) \in R$.
• Equivalence: Reflexive, symmetric, and transitive.

Function: A relation where each element of the domain is mapped to exactly one element in the codomain.

Types of Functions:

• Injective (One-to-one): Each output corresponds to one input.
• Surjective (Onto): Every element in the codomain is mapped to.
• Bijective: Both injective and surjective.

Example: Let $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^2$. Is $f$ surjective? Answer: No, because negative numbers are not in the range.

Related Question: Find the domain of $f(x) = \sqrt{x}$. Answer: $[0, \infty)$

---

4. Straight Lines

Rectangular Coordinate System: Points are defined by $(x, y)$ coordinates.

Distance Between Two Points:

$$ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Example: Distance between $(2,4)$ and $(-4,12)$:

$$ \sqrt{(-4-2)^2 + (12-4)^2} = \sqrt{36 + 64} = 10 $$

Section Formula: A point dividing a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in ratio $m:n$:

$$ \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) $$

Example: Midpoint of $(4,-2)$ and $(0,2)$:

$$ \left(\frac{4+0}{2}, \frac{-2+2}{2}\right) = (2, 0) $$

Area of a Triangle:

$$ \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| $$

Slope of a Line:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Equation of a Line:

• Slope-intercept: $y = mx + c$
• Point-slope: $y - y_1 = m(x - x_1)$
• Two-point: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

Parallel and Perpendicular Lines:

• Parallel: $m_1 = m_2$
• Perpendicular: $m_1 m_2 = -1$

Distance from a Point to a Line:

$$ \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} $$

Example: Find the distance of the line $3x + 4y - 5 = 0$ from $(1,2)$:

$$ \frac{|3(1) + 4(2) - 5|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 8 - 5|}{5} = \frac{6}{5} $$

Sum Squared Error (SSE):

$$ SSE = \sum_{i=1}^n (y_i - (mx_i + c))^2 $$

---

5. Quadratic Functions

Definition: $f(x) = ax^2 + bx + c, a \neq 0$

Axis of Symmetry: $x = -\frac{b}{2a}$

Vertex: $\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$

Types of Parabola:

• Upward: $a > 0$
• Downward: $a < 0$

Slope: $f’(x) = 2ax + b$

Quadratic Equation: $ax^2 + bx + c = 0$

Methods to Solve:

1. Factoring: $x^2 + 2x - 24 = 0 \implies (x+6)(x-4)=0 \implies x=-6, 4$
2. Completing the Square: $x^2 + 2x = 24 \implies (x+1)^2 = 25 \implies x = 4, -6$
3. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant: $D = b^2 - 4ac$

• $D > 0$: Two real roots
• $D = 0$: One real root
• $D < 0$: No real roots

Example: Solve $x^2 + 2x - 24 = 0$ Answer: $x = 4, -6$

---

6. Polynomial Functions

Definition: $f(x) = a_nx^n + \dots + a_0, a_n \neq 0$

Classification:

• By variables: Univariate, bivariate, multivariate
• By degree: Constant (0), linear (1), quadratic (2), cubic (3), quartic (4)
• By terms: Monomial (1), binomial (2), trinomial (3)

Operations:

• Addition: Combine like terms
• Subtraction: Similar to addition
• Multiplication: Distribute and combine like terms
• Division: Use long division or synthetic division

Example: Add $p(x) = x^3 + 3x^2 + 5x - 10$ and $q(x) = 3x^3 + 5x^2 - 6x - 20$ Answer: $4x^3 + 8x^2 - x - 30$

Zeroes: Values of $x$ for which $f(x) = 0$

---

7. Exponential and Logarithmic Functions

Exponential Function: $f(x) = a^x, a > 0, a \neq 1$

Laws of Exponents:

• $a^s \times a^t = a^{s+t}$
• $(a^s)^t = a^{st}$
• $(ab)^s = a^s b^s$

Graphing:

• For $a > 1$: Increasing
• For $0 < a < 1$: Decreasing

Natural Exponential: $f(x) = e^x$

Logarithmic Function: $f(x) = \log_a x$ is the inverse of $a^x$

Laws of Logarithms:

• $\log_a MN = \log_a M + \log_a N$
• $\log_a \frac{M}{N} = \log_a M - \log_a N$
• $\log_a M^r = r \log_a M$

Example: Solve $\log_8(x+1) + \log_8(x-1) = 1$ Answer: $\log_8((x+1)(x-1)) = 1 \implies x^2 - 1 = 8 \implies x = 3$

---

8. Composition and Inverse Functions

Composition: $(f \circ g)(x) = f(g(x))$

Example: $f(x) = 3x - 4, g(x) = x^2$ $(f \circ g)(x) = 3x^2 - 4$

Inverse Function: If $f$ is one-to-one, then $f^{-1}$ exists and $f^{-1}(f(x)) = x$

Example: $g(x) = 4x, h(x) = \frac{x}{4}$ $g$ is the inverse of $h$.

---

9. Exercises and Applications

Example: Solve $e^{-x^2} = \frac{(e^x)^2}{e^3}$ Answer: $e^{-x^2} = e^{2x-3} \implies -x^2 = 2x-3 \implies x^2 + 2x - 3 = 0 \implies x = 1, -3$

Example: If $N(t) = N_0 e^{-\lambda t}$, and at $t_1$, $N(t_1) = \frac{N_0}{2}$, at $t_2$, $N(t_2) = \frac{N_0}{4}$, what is the relation between $t_1$ and $t_2$? Answer: $t_2 = 2t_1$ (since half-life is constant)

---

Summary Table

Concept	Key Points & Formulas	Example/Question	Answer/Explanation
Set Theory	Subset, cardinality, set comprehension	Cardinality of ${a,b,c,d}$	4
Numbers	$\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$	Is $\frac{2}{4}$ rational?	Yes
Relations	Cartesian product, reflexive, symmetric, transitive	Is $R = {(1,1), (2,2)}$ reflexive?	Yes (if $S = {1,2}$)
Functions	Injective, surjective, bijective, domain, range	Domain of $\sqrt{x}$	$[0, \infty)$
Straight Lines	Distance, slope, equation, area	Distance $(2,4)$ to $(-4,12)$	10
Quadratic Functions	Vertex, axis, roots, discriminant	Solve $x^2 + 2x - 24 = 0$	$x = 4, -6$
Polynomials	Degree, terms, operations	Add $x^3 + 3x^2 + 5x - 10$ and $3x^3 + 5x^2 - 6x - 20$	$4x^3 + 8x^2 - x - 30$
Exponential/Logarithm	Laws, graphing, solving equations	Solve $\log_8(x+1) + \log_8(x-1) = 1$	$x = 3$
Composition/Inverse	$(f \circ g)(x) = f(g(x))$, $f^{-1}$	$f(x) = 3x - 4, g(x) = x^2$	$(f \circ g)(x) = 3x^2 - 4$

---

This structured approach covers all major concepts in the PDF, with examples and questions for clarity and practice¹.