Construction of Subsets and set operations

Construction of Subsets and set operations

A well-defined collection of distinct objects called elements or members.

Learning Outcomes:

  1. Write a set in its comprehensive form called set comprehension.
  2. Construct different subsets from a given set.
  3. Distinguish between closed interval and open interval.
  4. Demonstrate the ability to perform operations like union, intersection, set difference, and complement on sets using proper notation.

1️⃣ Set Comprehension: Writing Sets in Comprehensive Form

  • Set comprehension is a way to define a set by describing the properties its members must satisfy instead of listing elements.
  • Notation:
$$ A = \{x \mid \text{property of } x\} $$

Reads as: “Set $ A $ is the set of all $ x $ such that $ x $ satisfies the property.”

  • Example: Set of all even integers:
$$ E = \{x \mid x \in \mathbb{Z} \text{ and } x \text{ is even}\} $$

Venn diagrams illustrating set operations: Union, Intersection, Difference, and Symmetric Difference


2️⃣ Constructing Subsets from a Given Set

  • Given a set $ S = {a, b, c} $, subsets are any sets containing zero or more elements of $ S $.
  • All subsets of $ S $:
$$ \{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\} $$
  • Visualized as a power set containing $2^n$ subsets for a set of $n$ elements.
  • Diagram:

Power Set of {a,b,c}


3️⃣ Distinguish Between Closed Interval and Open Interval

  • Closed Interval $[a,b]$: Includes endpoints $a$ and $b$.
$$ [a, b] = \{x | a \leq x \leq b\} $$
  • Open Interval $(a,b)$: Excludes endpoints $a$ and $b$.
$$ (a, b) = \{x | a < x < b\} $$
  • Diagram:

Open vs Closed Interval

🟢 Solid dots indicate inclusion (closed), hollow dots indicate exclusion (open).


4️⃣ Set Operations: Union, Intersection, Set Difference, Complement

A graph illustrating set operations

  • Union ( $ A \cup B $ ): All elements in $ A $ or $ B $ or both. Example: $ A = {1,2}, B = {2,3} \Rightarrow A \cup B = {1,2,3} $
  • Intersection ( $ A \cap B $ ): Elements common to both $ A $ and $ B $. Example: $ A \cap B = {2} $
  • Set Difference ( $ A - B $ ): Elements in $ A $ but not in $ B $. Example: $ A - B = {1} $
  • Complement ( $ A^c $ or $ \bar{A} $ ): Elements not in $ A $ relative to the universal set $ U $.
  • Visual Diagram of Union and Intersection:

Union and Intersection

  • Visual Diagram of Set Difference and Complement:

Set Difference and Complement

Set Difference and Complement


Exercise Questions 🤯

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1) Which of the following is a correct representation of set comprehension?

A) $ {x \mid x \in \mathbb{N}, x is even} $ B) $ {x \in \mathbb{Z}, x is even} $ C) $ x is even, x \in \mathbb{N}, x } $ D) $ x is even, x \in \mathbb{Z}, x } $

Detailed Answer:

Proper set comprehension involves curly brackets and a “such that” ($\mid$ or “:”) description. A) is correct as it uses the proper notation:

$$ \{x \mid x \in \mathbb{N}, x \text{ is even}\} $$

B is missing the surrounding curly brackets and “such that”. C and D are not valid set notations.

Correct Answer: A

2) Which of the following is the set of natural numbers that are multiples of 3 or 5?

A) $ {x \mid x \in \mathbb{N}, x \bmod 3 = 0 and x \bmod 5 = 0} $ B) $ {x \mid x \in \mathbb{N}, x \bmod 3 = 0 or x \bmod 5 = 0} $ C) $ {x \mid x \in \mathbb{N}, x \bmod 3 = 0 and x \bmod 5 = 0} $ D) $ {x \mid x \in \mathbb{N}, x \bmod 3 = 0 or x \bmod 5 = 0} $

Detailed Answer:

Multiples of 3 or 5 are numbers divisible by either 3 or 5. So, “or” is appropriate, not “and”. B) and D) are identical and correct.

Correct Answer: B or D

3) Which of the following sets represents students from Madras University who play both cricket and basketball?

Options: A) {Abbas, Akhil, Rakesh} B) {Bhuvan, Charan, Dayal, Kiran} C) {Kiran, Charan, Rakesh} D) {Aditya, Jaivardhan, Nithin, Rakesh}

Detailed Answer:

Find students in both the cricket and basketball sets.

  • Basketball: {Abbas, Akhil, Aditya, Bhuvan, Charan, Dayal, Deepak, Kiran, Jaivardhan, Nithin, Rakesh}
  • Cricket: {Kiran, Charan, Rakesh, Kiran, Chidambaram, Balachandra}

Intersection = {Kiran, Charan, Rakesh}

Correct Answer: C

4) What is the number of students who play all the three sports?

Options: 1 2 3 4

Detailed Answer:

Find students who are in the intersection of all three sets.

Volleyball: {Akhil, Charan, Deepak, Sunil, Karthik, Rakesh} Basketball: {Abbas, Akhil, Aditya, Bhuvan, Charan, Dayal, Deepak, Kiran, Jaivardhan, Nithin, Rakesh} Cricket: {Kiran, Charan, Rakesh, Kiran, Chidambaram, Balachandra}

Students present in all three: Check each name: Charan and Rakesh only.

So, number is 2.

Correct Answer: 2

5) Which of the following set represents the students from Madras University who play volleyball but not cricket?

Options: A) {Akhil, Rakesh, Sunil, Karthik} B) {Sunil, Karthik} C) {Deepak, Sunil, Bhuvan, Dayal, Nithin} D) {Abbas, Aditya, Bhuvan, Dayal, Kiran, Jaivardhan, Nithin}

Detailed Answer:

Volleyball: {Akhil, Charan, Deepak, Sunil, Karthik, Rakesh} Cricket: {Kiran, Charan, Rakesh, Kiran, Chidambaram, Balachandra}

Volleyball ∖ Cricket: Remove Charan and Rakesh from volleyball set. {Akhil, Deepak, Sunil, Karthik}

Correct Answer: A

6) Which of the following sets represent the students from Madras University who are not playing cricket?

Options: A) {Dayal, Deepak, Kiran, Jaivardhan} B) {Rakesh, Kiran, Chidambaram, Balachandra} C) {Bhuvan, Jaivardhan, Sunil, Chidambaram} D) {Sunil, Chidambaram, Karthik, Balachandra}

Detailed Answer:

Find students not in the cricket set.

Cricket: {Kiran, Charan, Rakesh, Kiran, Chidambaram, Balachandra}

Check who is not listed in this set. From the student list, the only correct option is D.

Correct Answer: D

7) If exams A and B are going to be conducted simultaneously, then what are the items that are required to be carried by the student to the exam centre?

Given sets: A: {Hall ticket, Pen, Calculator, Ruler, Pencil} B: {Compass, Pencil, Drafter, Ruler, Eraser, Sharpener, Hall ticket}

Options: A) {Hall ticket, Ruler, Pencil, Eraser, Sharpener, Pen, Drafter, Compass, Calculator} B) {Calculator, Ruler, Pencil, Hall ticket, Pen, Paper, Compass, Drafter} C) {Pen, Calculator, Pencil, Hall ticket, Ruler, Compass, Drafter, Eraser} D) {Ruler, Pencil, Compass, Drafter, Eraser, Sharpener, Hall ticket, Calculator}

Detailed Answer:

The student needs all items from both sets (union):

{Hall ticket, Pen, Calculator, Ruler, Pencil, Compass, Drafter, Eraser, Sharpener}

Option A covers all these exactly.

Correct Answer: A

8) If a student forgets to bring her Calculator, which exam will she not be allowed to write?

Options: A) Exam B B) She can write both exams C) Exam A D) None of the above

Detailed Answer:

Calculator is required only for Exam A (not listed for B).

Correct Answer: C