Unit 1: Set Theory, Relation and Function - Subjective Questions

MTH136 — Discrete Structures • Practice Questions with Detailed Answers

20 questions

Define a set and explain the different methods used to describe a set with suitable examples.

Explain the various types of sets with definitions and examples.

Define subset and power set. Find the power set of $A = \{a, b, c\}$ and state the relation between the cardinality of a set and its power set.

Describe Venn diagrams and illustrate the operations of union, intersection, difference, and complement using Venn diagrams.

State and prove De Morgan's Laws for sets, and verify them using a Venn diagram description.

State the laws of set theory (algebra of sets) with their formulas.

Prove the Distributive Law $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ .

Define the Cartesian product of two sets. If $A = \{1, 2\}$ and $B = \{x, y, z\}$ , find $A \times B$ and $B \times A$ . Is $A \times B = B \times A$ ?

Define a relation. Explain the different types of relations with examples.

Let $R$ be a relation on the set $A = \{1, 2, 3\}$ defined by $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$ . Check whether $R$ is reflexive, symmetric, and transitive.

Define a function. Distinguish between a relation and a function with examples.

Function: A function $f$ from set $A$ to set $B$ (written $f : A \to B$ ) is a special relation in which every element of $A$ is associated with exactly one element of $B$ .

$A$ is the domain, $B$ is the codomain.
The set of all outputs is the range, where $\text{Range} \subseteq B$ .

Difference between Relation and Function:

Basis	Relation	Function
Definition	Any subset of $A \times B$	Special relation
Mapping	An element of $A$ may map to zero, one, or many elements of $B$	Each element of $A$ maps to exactly one element of $B$
Uniqueness	Not required	Required

Example of Relation (not a function):
$R = \{(1,a), (1,b), (2,c)\}$ — here $1$ maps to two values, so not a function.

Example of Function:
$f = \{(1,a), (2,b), (3,c)\}$ — each input has a unique output.

Explain the identity function and the polynomial function along with their graphs and properties.

Define the modulus (absolute value) function. Describe its properties and sketch its graph.

Explain the greatest integer function (step function). Evaluate $\lfloor 3.7 \rfloor$ , $\lfloor -2.3 \rfloor$ , and $\lfloor 5 \rfloor$ , and describe its graph.

Distinguish between one-one (injective) and onto (surjective) functions with definitions and examples.

One-One (Injective) Function:
A function $f : A \to B$ is one-one if different elements of $A$ have different images in $B$ .
$f(a_1) = f(a_2) \Rightarrow a_1 = a_2$
Equivalently, $a_1 \neq a_2 \Rightarrow f(a_1) \neq f(a_2)$ .

Example: $f(x) = 2x + 1$ on $\mathbb{R}$ is one-one.

Onto (Surjective) Function:
A function $f : A \to B$ is onto if every element of the codomain $B$ has at least one pre-image in $A$ , i.e. $\text{Range} = \text{Codomain}$ .
$\forall\, y \in B, \ \exists\, x \in A \text{ such that } f(x) = y$

Example: $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x^3$ is onto.

Key Differences:

Basis	One-One	Onto
Focus	Distinct inputs give distinct outputs	Every output is achieved
Condition	$f(a_1)=f(a_2)\Rightarrow a_1=a_2$	Range = Codomain

Bijective Function: A function that is both one-one and onto.

Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x - 5$ is both one-one and onto (bijective).

In a survey of $100$ students, $60$ like tea, $50$ like coffee, and $30$ like both. Using set theory, find how many students like (i) at least one drink, (ii) only tea, (iii) neither drink.

Prove the Associative Law of Union $(A \cup B) \cup C = A \cup (B \cup C)$ and verify it using the sets $A = \{1,2\}$ , $B = \{2,3\}$ , $C = \{3,4\}$ .

Given $A = \{1,2,3,4,5\}$ , $B = \{2,4,6,8\}$ with universal set $U = \{1,2,\dots,10\}$ , compute $A \cup B$ , $A \cap B$ , $A - B$ , $B - A$ , and $A'$ .

Explain the operations of union, intersection, and difference of sets with definitions, notations, and examples. Also state whether each operation is commutative.

Question 1

Define a set and explain the different methods used to describe a set with suitable examples.

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Answer 1

A set is a well-defined collection of distinct objects, called elements or members of the set. The word well-defined means that it must be possible to decide clearly whether a given object belongs to the set or not.

Methods of describing a set:

Roster / Tabular Form: All elements are listed explicitly within braces, separated by commas.
- Example: $A = \{1, 2, 3, 4, 5\}$
Set-Builder / Rule Form: The elements are described by a common property.
- Example: $A = \{x : x \in \mathbb{N}, x \leq 5\}$

Notation:

$x \in A$ means $x$ is an element of $A$ .
$x \notin A$ means $x$ is not an element of $A$ .

Example: The set of vowels can be written as:

Roster form: $V = \{a, e, i, o, u\}$
Set-builder form: $V = \{x : x \text{ is a vowel in the English alphabet}\}$

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Question 2

Explain the various types of sets with definitions and examples.

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Answer 2

Different types of sets are classified based on the number of elements and their nature:

Empty / Null Set: A set with no elements, denoted by or .
- Example: $\{x : x^2 = -1, x \in \mathbb{R}\} = \emptyset$
Singleton Set: A set with exactly one element.
- Example: $\{5\}$
Finite Set: A set with a countable (finite) number of elements.
- Example: $\{1, 2, 3\}$
Infinite Set: A set with an uncountable / unending number of elements.
- Example: $\mathbb{N} = \{1, 2, 3, \dots\}$
Equal Sets: Two sets having exactly the same elements. .
- Example: $\{1,2,3\} = \{3,2,1\}$
Equivalent Sets: Two sets having the same number of elements (same cardinality).
- Example: $\{a,b,c\}$ and $\{1,2,3\}$
Universal Set: The set containing all objects under consideration, denoted by $U$ .
Subset: $A \subseteq B$ if every element of $A$ is in $B$ .
Disjoint Sets: Sets having no common elements, i.e. $A \cap B = \emptyset$ .

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Question 4

Describe Venn diagrams and illustrate the operations of union, intersection, difference, and complement using Venn diagrams.

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Answer 4

Venn Diagrams are pictorial representations of sets using closed figures (usually circles) drawn inside a rectangle that represents the universal set $U$ . They help visualize relationships and operations between sets.

Set Operations illustrated:

Union ( $A \cup B$ ): All elements in $A$ , $B$ , or both. Both circles are shaded fully.
$A \cup B = \{x : x \in A \text{ or } x \in B\}$
Intersection ( $A \cap B$ ): Only the common (overlapping) region is shaded.
$A \cap B = \{x : x \in A \text{ and } x \in B\}$
Difference ( $A - B$ ): The part of $A$ outside $B$ is shaded.
$A - B = \{x : x \in A \text{ and } x \notin B\}$
Complement ( $A'$ ): The region inside $U$ but outside $A$ is shaded.
$A' = \{x : x \in U \text{ and } x \notin A\}$

Venn diagrams are widely used to verify set identities such as De Morgan's laws visually.

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Question 9

Define a relation. Explain the different types of relations with examples.

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Answer 9

Relation: A relation $R$ from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$ . If $(a,b) \in R$ , we write $a\,R\,b$ .

A relation on a set $A$ is a subset of $A \times A$ .

Types of Relations on a set $A$ :

Reflexive: for every .
- Example: $\leq$ on real numbers.
Symmetric: If then .
- Example: "is a sibling of".
Transitive: If and , then .
- Example: $<$ on integers.
Anti-symmetric: If and , then .
- Example: $\leq$ on integers.
Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
- Example: "is equal to", congruence modulo $n$ .
Empty Relation: $R = \emptyset$ .
Universal Relation: $R = A \times A$ .

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Question 11

Define a function. Distinguish between a relation and a function with examples.

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Answer 11

Function: A function $f$ from set $A$ to set $B$ (written $f : A \to B$ ) is a special relation in which every element of $A$ is associated with exactly one element of $B$ .

$A$ is the domain, $B$ is the codomain.
The set of all outputs is the range, where $\text{Range} \subseteq B$ .

Difference between Relation and Function:

Basis	Relation	Function
Definition	Any subset of $A \times B$	Special relation
Mapping	An element of $A$ may map to zero, one, or many elements of $B$	Each element of $A$ maps to exactly one element of $B$
Uniqueness	Not required	Required

Example of Relation (not a function):
$R = \{(1,a), (1,b), (2,c)\}$ — here $1$ maps to two values, so not a function.

Example of Function:
$f = \{(1,a), (2,b), (3,c)\}$ — each input has a unique output.

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Question 12

Explain the identity function and the polynomial function along with their graphs and properties.

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Answer 12

Identity Function:
The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x$ is called the identity function.

Domain: $\mathbb{R}$ , Range: $\mathbb{R}$
Graph: A straight line passing through the origin with slope $1$ (the line $y = x$ ).
It is both one-one and onto (bijective).

Polynomial Function:
A function of the form:
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$
where $a_0, a_1, \dots, a_n$ are real constants ( $a_n \neq 0$ ) and $n$ is a non-negative integer, is a polynomial function of degree $n$ .

Examples:
- $f(x) = 3$ (constant, degree 0)
- $f(x) = 2x + 1$ (linear, degree 1)
- $f(x) = x^2 - 4$ (quadratic, degree 2)
Graphs:
- Linear functions give straight lines.
- Quadratic functions give parabolas.
Domain: $\mathbb{R}$ for all polynomial functions.

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Question 15

Distinguish between one-one (injective) and onto (surjective) functions with definitions and examples.

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Answer 15

Example: $f(x) = 2x + 1$ on $\mathbb{R}$ is one-one.

Example: $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x^3$ is onto.

Key Differences:

Basis	One-One	Onto
Focus	Distinct inputs give distinct outputs	Every output is achieved
Condition	$f(a_1)=f(a_2)\Rightarrow a_1=a_2$	Range = Codomain

Bijective Function: A function that is both one-one and onto.

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Unit2

Unit 1: Set Theory, Relation and Function - Subjective Questions

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