Week 1: Introduction to Sets, Relations, and Functions

Course: Jan 2026 - Mathematics I Difficulty: Foundational Focus: Abstract structures, basic mappings, and notation.

1. Sets and Set Theory

1.1 Representation of Sets

1. Roster (Tabular) Form: Listing all elements separated by commas, enclosed in curly braces. Example: $V = \{a, e, i, o, u\}$.
2. Set-Builder Form: Stating a property that all elements of the set must satisfy. Example: $V = \{x \mid x \text{ is a vowel in the English alphabet}\}$.

1.2 Types of Sets

• Empty Set (Null Set/Void Set): A set containing no elements. Denoted by $\emptyset$ or $\{\}$.
• Singleton Set: A set containing exactly one element. Example: $\{5\}$.
• Finite and Infinite Sets: A set is finite if the process of counting its elements terminates. Otherwise, it is infinite. Example of infinite set: Natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$.
• Equal Sets: Two sets are equal if they have exactly the same elements.
• Subset: Set $A$ is a subset of set $B$ ($A \subseteq B$) if every element of $A$ is also an element of $B$.
• Power Set: The set of all subsets of a set $A$ is denoted by $\mathcal{P}(A)$. If $A$ has $n$ elements, $\mathcal{P}(A)$ has $2^n$ elements.

1.3 Set Operations

• Union ($A \cup B$): The set of all elements which are in $A$, in $B$, or in both.
• Intersection ($A \cap B$): The set of all elements which are common to both $A$ and $B$.
• Difference ($A - B$): The set of elements which belong to $A$ but not to $B$.
• Complement ($A'$ or $A^c$): With respect to a universal set $U$, it is the set of all elements in $U$ that are not in $A$.

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2. Cartesian Product and Relations

2.1 Cartesian Product

2.2 Relations

• Domain: The set of all first elements of the ordered pairs in a relation $R$.
• Range: The set of all second elements of the ordered pairs in a relation $R$.
• Codomain: The entire set $B$ is called the codomain of the relation $R$. Note that Range $\subseteq$ Codomain.

2.3 Types of Relations (on a single set A)

1. Reflexive: If $(a, a) \in R$ for every $a \in A$.
2. Symmetric: If $(a, b) \in R$ implies that $(b, a) \in R$ for all $a, b \in A$.
3. Transitive: If $(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R$ for all $a, b, c \in A$.

3. Functions

• The element $y$ is called the image of $x$ under $f$, and $x$ is called the pre-image of $y$.
• Domain: The set $A$. (Every element in $A$ must be mapped).
• Codomain: The set $B$.
• Range: The set of all images of elements of $A$. (Range $\subseteq$ Codomain).

3.1 Classifications of Functions

• Injective (One-to-One): Distinct elements of $A$ have distinct images in $B$. That is, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.
• Surjective (Onto): Every element of $B$ is the image of some element of $A$. That is, the Range is equal to the Codomain.
• Bijective (One-to-One and Onto): A function that is both injective and surjective. Bijective functions are uniquely invertible.

3.2 Verification

• Vertical Line Test: A curve drawn in a plane represents a function if and only if no vertical line intersects the curve more than once.
• Horizontal Line Test: Used to determine if a function is injective. If any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

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