Sets and Functions

What is a Set?

A set is simply a well-defined collection of distinct objects. "Well-defined" means there should be no confusion about whether an object belongs to the set or not.

Example: "The set of students taller than 5'5"" is a set. However, "The set of all tall students" is NOT a set because "tall" is subjective.

Representation of Sets

• Roster (Tabular) Form: Listing every element inside curly brackets.
  Example: $V = \{a, e, i, o, u\}$
• Set-Builder Form: Describing the common property of the elements.
  Example: $V = \{x : x \text{ is a vowel in the English alphabet}\}$

Types of Sets

• Empty (Null) Set: No elements. Denoted by $\emptyset$ or $\{ \}$.
• Finite and Infinite Sets: Finite are countable (e.g., days of week); Infinite go on forever (e.g., natural numbers).
• Equal Sets: Sets with exactly the same elements.
• Subset: If every element of A is also in B, then $A \subseteq B$.
• Power Set: Set of all possible subsets. For $n$ elements, the power set has $2^n$ elements.
• Universal Set ($U$): The set containing all objects under consideration.

Examples

• If $A = \{1, 2\}$, its subsets are $\emptyset, \{1\}, \{2\}, \{1, 2\}$.
• The set of all even prime numbers is $\{2\}$ (Singleton set).

Real-Life Use: Digital File Management

Think of your computer's folder system. A "Folder" is a Set. "Photos" is a set, "Vacation 2023" is a subset, and the "Hard Drive" is the Universal Set.

Set Operations

• Union ($\cup$): Combining everything from both sets ("A or B").
• Intersection ($\cap$): Only common elements ("A and B").
• Difference (A - B): Elements in A that are NOT in B.
• Complement ($A'$): Elements in the Universal set not in set A.

Examples

If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$:

• $A \cup B = \{1, 2, 3, 4, 5\}$
• $A \cap B = \{3\}$

Venn Diagrams

Visual representations where the Universal set is a rectangle and sets are circles. Overlapping areas represent the intersection.

Real-Life Use: E-commerce Filters

When filtering for "Electronics" AND "Price under $500" AND "4 Stars," the website performs an Intersection operation.

Cartesian Product

The Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

Example: If $A = \{1, 2\}$ and $B = \{x, y\}$, then $A \times B = \{(1, x), (1, y), (2, x), (2, y)\}$.

What is a Relation?

A relation is a subset of the Cartesian product that defines a specific connection between two sets.

Types of Relations

• Reflexive: Every element relates to itself: $(a, a) \in R$.
• Symmetric: If $(a, b) \in R$, then $(b, a) \in R$.
• Transitive: If $a \to b$ and $b \to c$, then $a \to c$.
• Equivalence Relation: A relation that is Reflexive, Symmetric, AND Transitive.

Real-Life Use: Social Media Networks

Facebook "Friendship" is a Symmetric Relation. Instagram "Following" is NOT symmetric (one-way following).

Meaning and Definition

A function is a relation where every element in the Domain is mapped to exactly one element in the Codomain.

Rule: One input cannot have two different outputs.

Key Terms

• Domain: All possible inputs.
• Codomain: All potential outputs.
• Range: The actual outputs that are hit by the function.

Types of Functions

• One-to-One (Injective): Distinct inputs give distinct outputs.
• Onto (Surjective): Every element in codomain is matched (Range = Codomain).
• Bijective: Both One-to-One and Onto. These are invertible.

Real-Life Use: Vending Machines

Pressing "A1" (Input) gives you one specific snack (Output). If "A1" gave two different items randomly, it would not be a function!

The Big Three: $\sin x, \cos x, \tan x$

Graphs Summary

• $\sin x$: Wave starting at $(0,0)$.
• $\cos x$: Wave starting at $(0,1)$.
• $\tan x$: Vertical curves with asymptotes.

Real-Life Use: Noise Cancelling Headphones

Headphones analyze incoming noise (sine wave) and create an inverted sine function to cancel sound through destructive interference.

Relation vs. Function

Essential Formulas

• Subsets: $2^n$
• De Morgan's Laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$
• Cardinality: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

• SQL Databases: Joins (Inner, Left) are based on Set Theory intersections.
• Machine Learning: Activation Functions (ReLU, Sigmoid) determine neuron firing.
• Cryptography: RSA encryption uses bijective functions and modular arithmetic.

Scenario: QuickShip Logistics uses Sets, Relations, and Functions to manage packages.

• Sets: Intersecting "Heavy Packages" and "Express Shipping" sets helps allocate larger vehicles.
• Relations: Mapping Drivers to Locations. Route optimization checks for the "Transitive" property.
• Functions: A Pricing Function $f(w) = 5 + 2w$ ensures consistent billing based on weight.

1. If set $A = \{1, 2, 3\}$, how many subsets does it have?
   Ans: (c) 8
2. A set with no elements is called:
   Ans: (b) Null set
3. If $A \cap B = \emptyset$, the sets are called:
   Ans: (c) Disjoint
4. The set of all possible subsets is the:
   Ans: (b) Power set
5. A relation $R$ is symmetric if $(a, b) \in R$ implies:
   Ans: (b) $(b, a) \in R$
6. A function where every element in the codomain has a pre-image is:
   Ans: (b) Surjective
7. What is the domain of $\tan x$?
   Ans: (c) $\mathbb{R} - \{(2n+1)\pi/2\}$
8. The range of $\sin x$ is:
   Ans: (b) $[-1, 1]$
9. If $f: A \to B$ is both one-to-one and onto, it is called:
   Ans: (c) Bijective
10. Which of these is NOT a function?
    Ans: (b) $\{(1, 2), (1, 3)\}$

1. Q: Explain the difference between a Subset and a Power Set.
   Ans: A subset is a set whose elements are contained within another. A Power Set is the set of ALL possible subsets. Example: $S=\{1,2\} \implies P(S)=\{\emptyset, \{1\}, \{2\}, \{1,2\}\}$.
2. Q: Prove $(A \cup B)' = A' \cap B'$.
   Ans: $A \cup B$ is the union of two circles. Its complement is everything outside both. This area is where the complement of A and the complement of B overlap (intersection).
3. Q: Three conditions for an Equivalence Relation?
   Ans: 1. Reflexive, 2. Symmetric, 3. Transitive.
4. Q: Codomain vs. Range?
   Ans: Codomain is the set of all potential outputs; Range is the set of outputs that actually occur.
5. Q: When is $\tan x$ undefined?
   Ans: When $\cos x = 0$, which occurs at $x = (2n+1)\pi/2$.
6. Q: Is $f(x) = x^3$ bijective?
   Ans: Yes, it is one-to-one (distinct $x$ gives distinct $x^3$) and onto (every real number has a cube root).
7. Q: If $n(A)=20, n(B)=30, n(A \cup B)=45$, find $n(A \cap B)$.
   Ans: $45 = 20 + 30 - n(A \cap B) \implies n(A \cap B) = 5$.
8. Q: Describe the graph of $\cos x$.
   Ans: A periodic wave starting at $(0,1)$, dipping to $-1$ at $\pi$, and returning to $1$ at $2\pi$.
9. Q: What is a constant function?
   Ans: A function where every input maps to the same output. Example: $f(x)=5$.
10. Q: If $A=\{1,2\}$ and $B=\{3,4\}$, find $A \times B$.
    Ans: $\{(1,3), (1,4), (2,3), (2,4)\}$.

• Relation Types (RST): Think "Real Super Teams" (Reflexive, Symmetric, Transitive).
• Trig Signs (CAST Rule): All Silver Tea Cups (All positive $\to$ Sin positive $\to$ Tan positive $\to$ Cos positive).
• Formula: Number of subsets = $2^n$.

Mnemonic: For an Equivalence relation, remember: "Self-love (Reflexive), Mirror (Symmetric), and Chain Reaction (Transitive)."