Concepts & Solved Examples
Learn subsets, proper subsets, set cardinality notations $n(A)$, power sets $P(A)$, and universal sets $U$.
Exercise 1.2 Solutions
Detailed step-by-step solutions to all parts of Exercise 1.2 with interactive solutions.
Chapter Worksheets
Three levels (Easy, Medium, Hard) of custom worksheets designed to perfect your power set skills.
Revision Notes
Quick formulas, notation keys, common mistakes, and key takeaways for exam revision.
Interactive Tests
Basic, Standard, and Advanced quizzes with 10 questions each to test your understanding of subsets.
What You'll Learn in This Section
In this section, we study how sets relate to one another as subsets and proper subsets. We introduce mathematical cardinality to measure set sizes, define power sets that compile all possible subsets, and establish the universal bounding set.
• Subsets ($\subseteq$): Understanding containment relationships and proper subsets ($\subsetneq$).
• Empty Set Subset Rule: Exploring why $\emptyset$ is a subset of every set.
• Cardinality ($n(A)$): Finding the number of distinct elements in finite sets.
• Power Sets ($P(A)$): Listing all subsets and calculating subset counts using $2^p$.
• Universal Sets ($U$): Framing contexts with bounding sets.
Start Learning
Click on Concepts & Solved Examples to begin studying the theory, or go straight to Exercise 1.2 Solutions to check your homework answers.
Subsets, Cardinality & Power Sets: Concepts
Concept of Subsets
Consider the set $X$ of students who live within a 5 km radius around your school. These students obviously, along with others, belong to the set $Y$ of all students in your class.
Since every student of set $X$ also belongs to (or is contained in) set $Y$, we say that $X$ is a subset of $Y$.
A set $A$ is said to be a subset of $B$ if and only if every element of $A$ is also an element of $B$. We write this symbolically as:
$A \subseteq B$ (read as "$A$ is a subset of $B$").
If $A$ is not a subset of $B$, we write $A \not\subseteq B$. Formally, $A \subseteq B \iff (a \in A \Rightarrow a \in B)$.
Proper Subsets
If $A \subseteq B$ and $A \neq B$ (meaning $B$ has at least one element not present in $A$), we say that $A$ is a proper subset of $B$. Symbolically, we write:
$A \subsetneq B$ (or in some texts simply $A \subset B$).
Is the empty set $\emptyset$ a subset of a set $P$ containing at least one element? Yes! The empty set $\emptyset$ has no elements, so the condition "$a \in \emptyset \Rightarrow a \in P$" holds vacuously. The empty set $\emptyset$ is a subset of every set.
(i) $A = \{2, 3, p\}$ and $B = \{1, 2, 3, p, q\}$
(ii) $A = \{2, 3, 5, 7\}$ and $B = \emptyset$
(iii) $A = \{3, 8, 9, 0\}$ and $B = \{0, 9, 8, 3\}$
(iv) $A = \{x \mid x = 2n, n \in \mathbb{N}\}$ and $B = \{x \mid x = 4n, n \in \mathbb{N}\}$
Show Solution
(ii) Since $\emptyset$ is a subset of every set, we have $B \subseteq A$.
(iii) Since $A$ and $B$ have the exact same elements, they are equal. Thus, both $A \subseteq B$ and $B \subseteq A$ are true.
(iv) $A = \{2, 4, 6, 8, 10, 12, \dots\}$ (multiples of 2). $B = \{4, 8, 12, 16, \dots\}$ (multiples of 4). Every element in $B$ is also in $A$. Thus, $B \subseteq A$.
Cardinality of a Set
The cardinality of a set describes its size by counting its unique elements.
If there are exactly $m$ distinct elements in a set $A$, we say the cardinality of set $A$ is $m$. Symbolically, we write it as:
$n(A) = m$, where $m$ is a non-negative integer.
For example:
- If $A = \{1, 2, 3, \dots, 9\}$, then $n(A) = 9$.
- If $B = \left\{\frac{1}{2}, 3, 7, 0\right\}$, then $n(B) = 4$.
- For the empty set, $n(\emptyset) = 0$.
Power Sets
Consider the set $D = \{a, b\}$. Let's write down all of its subsets. Subsets can have 0 elements, 1 element, or 2 elements:
- 0-element subsets: $\emptyset$
- 1-element subsets: $\{a\}, \{b\}$
- 2-element subsets: $\{a, b\}$
Thus, $D$ has exactly 4 subsets: $\emptyset, \{a\}, \{b\}, \{a, b\}$. If we group all these subsets into a single set, we get a power set.
The collection of all subsets of a set $A$ is called the power set of $A$, denoted by $P(A)$.
In $P(D)$: the objects $\emptyset$, $\{a\}$, $\{b\}$, and $\{a, b\}$ are **elements** of $P(D)$. Therefore:
• $\{a\} \in P(D)$ is TRUE.
• $a \in P(D)$ is FALSE (since $a$ is an element of $D$, but not of $P(D)$).
Show Solution
- 0-element subsets: $\emptyset$
- 1-element subsets: $\{0\}, \{1\}, \{2\}$
- 2-element subsets: $\{0, 1\}, \{0, 2\}, \{1, 2\}$
- 3-element subsets: $\{0, 1, 2\}$
$\Rightarrow P(A) = \{\emptyset, \{0\}, \{1\}, \{2\}, \{0, 1\}, \{0, 2\}, \{1, 2\}, \{0, 1, 2\}\}$.
Show Solution
$P(\emptyset) = \{\emptyset\}$ (which contains exactly 1 element).
2. For the set $\{\emptyset\}$ (which has 1 element, namely $\emptyset$): its subsets are $\emptyset$ and $\{\emptyset\}$ itself. Therefore:
$P(\{\emptyset\}) = \{\emptyset, \{\emptyset\}\}$ (which contains exactly 2 elements).
Formula for Number of Subsets
If a set $A$ has $p$ elements, i.e., $n(A) = p$, then the number of subsets of $A$ is given by:
The number of proper subsets is $2^p - 1$ (excluding the set itself).
Universal Set
Usually, in any discussion, we choose a fixed large set containing all the elements of various subsets under consideration. This large set is called the Universal Set and is denoted by $U$.
Examples of Universal Sets:
- If we are studying integers $\mathbb{Z}$, the universal set could be the set of rational numbers $\mathbb{Q}$ or real numbers $\mathbb{R}$.
- If we are analyzing the sets of students in the math club and science club, the universal set $U$ could be the set of all students in the school.
Exercise 1.2 — Ideal Textbook Solutions
Question 1
Fill in the blanks with the symbol $\subseteq$ or $\not\subseteq$
(i) $\{2, 3, 4\}$ ........ $\{1, 2, 3, 4, 5\}$
(ii) $\{x \mid x \text{ are triangles in a plane}\}$ ........ $\{x \mid x \text{ are polygons in a plane}\}$
(iii) $\{x \text{ is an integer}\}$ ........ $\{x \mid x \text{ is a multiple of } 4\}$
(iv) $\emptyset$ ........ $\{\}$
(v) $\left\{x \mid x = \frac{1}{m} \text{ where } m \text{ is a non-zero integer}\right\}$ ........ $\{x \mid x \text{ is a rational number}\}$
(vi) $\{x \mid x = n^2\}$ ........ $\{x \mid x = n^3\}$, (where $n$ is a natural number)
(vii) $\{x \mid x \in \mathbb{R}\}$ ........ $\{x \mid x = 2^n\}$, (where $n$ is a natural number/integer)
View Complete Solution & Explanation
(i) Solution: $\subseteq$
Every element ($2, 3, 4$) is present in the second set. Thus, it is a subset.
(ii) Solution: $\subseteq$
A triangle is a 3-sided polygon. Thus, every triangle is a polygon, making the set of triangles a subset of polygons.
(iii) Solution: $\not\subseteq$
An integer (e.g., $1, 2, 3$) is not necessarily a multiple of 4. In fact, the relationship is the opposite: the multiples of 4 are a subset of integers, not vice-versa.
(iv) Solution: $\subseteq$ (or even $=$)
Both notations represent the empty set. Since every set is a subset of itself, $\emptyset \subseteq \{\}$ is true.
(v) Solution: $\subseteq$
Since $m$ is a non-zero integer, $\frac{1}{m}$ is a division of integers, which is by definition a rational number. Thus, every element of the first set is in the second set.
(vi) Solution: $\not\subseteq$
The square set is $\{1, 4, 9, 16, 25, \dots\}$. The cube set is $\{1, 8, 27, 64, \dots\}$. The number $4$ is in the first set but not in the second, so it is not a subset.
(vii) Solution: $\not\subseteq$
The set of real numbers $\mathbb{R}$ contains numbers like $0.5, 3.14$, which cannot be written in the form $2^n$ (where $n$ is a natural number). Thus, $\mathbb{R}$ is not a subset of $\{x \mid x = 2^n\}$.
Question 2
True or False Statements
Determine whether the following statements are true or false:
(i) $1 \subseteq \{1\}$
(ii) $\{2\} \subseteq \{2\}$
(iii) $\{2\} \in \{\{2\}\}$
(iv) $\emptyset \subseteq \{1, 2, 3\}$
(v) $\emptyset \in \{1, 2, 3\}$
View Complete Solution & Explanation
(i) Solution: False
The symbol $\subseteq$ denotes a relationship between two sets. The number $1$ is an element, not a set. The correct statement would be $1 \in \{1\}$ or $\{1\} \subseteq \{1\}$.
(ii) Solution: True
Every set is a subset of itself. Since all elements in $\{2\}$ belong to $\{2\}$, $\{2\} \subseteq \{2\}$ holds true.
(iii) Solution: True
The set $\{\{2\}\}$ is a nested set whose sole element is $\{2\}$. Since $\{2\}$ is listed directly inside, $\{2\} \in \{\{2\}\}$ is true.
(iv) Solution: True
The empty set $\emptyset$ is a subset of every set. Since the condition holds vacuously, $\emptyset \subseteq \{1, 2, 3\}$ is true.
(v) Solution: False
The empty set $\emptyset$ is a subset of $\{1, 2, 3\}$, but it is not an element of it. The elements of $\{1, 2, 3\}$ are simply $1, 2, 3$, none of which are $\emptyset$.
Question 3
Write the Power Set
Write the power set $P(S)$ of the following sets:
(i) $\{1\}$ (ii) $\{p, q\}$ (iii) $\{1, 2, 5\}$ (iv) $\{\emptyset, \{\}\}$
View Complete Solution & Explanation
(i) Solution:
Subsets of $\{1\}$ are $\emptyset$ and $\{1\}$.
$P(S) = \{\emptyset, \{1\}\}$
(ii) Solution:
Subsets of $\{p, q\}$ are $\emptyset, \{p\}, \{q\}, \{p, q\}$.
$P(S) = \{\emptyset, \{p\}, \{q\}, \{p, q\}\}$
(iii) Solution:
Subsets of $\{1, 2, 5\}$ are $\emptyset$, $\{1\}$, $\{2\}$, $\{5\}$, $\{1, 2\}$, $\{1, 5\}$, $\{2, 5\}$, $\{1, 2, 5\}$.
$P(S) = \{\emptyset, \{1\}, \{2\}, \{5\}, \{1, 2\}, \{1, 5\}, \{2, 5\}, \{1, 2, 5\}\}$
(iv) Solution:
Note that $\emptyset$ and $\{\}$ are the exact same empty set. Since a set cannot contain repeated identical elements, the set $\{\emptyset, \{\}\}$ simplifies to $\{\emptyset\}$ (which has exactly 1 element, which is the empty set itself).
Its subsets are $\emptyset$ and $\{\emptyset\}$. Therefore:
$P(S) = \{\emptyset, \{\emptyset\}\}$
Question 4
Cardinality of Sets
What is the cardinality of the following sets:
(i) $\{a\}$
(ii) $\{a, \{a\}\}$
(iii) $\{\emptyset, 1, 2, \{1, 2\}\}$
(iv) $\{1, \{1\}, \{1, \{1\}\}\}$
(v) $\{\emptyset, \{\}, \{\emptyset, \{\}\}\}$
View Complete Solution & Explanation
(i) Solution: 1
There is only 1 element, which is $a$.
(ii) Solution: 2
There are 2 distinct elements: the element $a$, and the set $\{a\}$.
(iii) Solution: 4
The distinct elements are: $\emptyset$, $1$, $2$, and the set $\{1, 2\}$. Total count = 4.
(iv) Solution: 3
The distinct elements are: $1$, $\{1\}$, and $\{1, \{1\}\}$. Total count = 3.
(v) Solution: 2
Let's simplify the set: since $\emptyset$ and $\{\}$ are identical, we write them once. The third element is $\{\emptyset, \{\}\}$, which simplifies to $\{\emptyset\}$.
The set can be written as $\{\emptyset, \{\emptyset\}\}$. This has exactly 2 elements: the empty set $\emptyset$, and the set containing the empty set $\{\emptyset\}$. Cardinality is 2.
Question 5
Subset & Power Set Cardinality Calculations
Let $A$ be a set and $n(A) = 10$, then find the value of $n[P(A)]$. What if $A$ has $100$ elements?
View Complete Solution & Explanation
We know that if a set $A$ has $p$ elements, then its power set $P(A)$ contains $2^p$ elements. Formally, $n[P(A)] = 2^{n(A)}$.
1. If $n(A) = 10$:
$n[P(A)] = 2^{10} = \mathbf{1024}$ elements.
2. If $n(A) = 100$:
$n[P(A)] = \mathbf{2^{100}}$ elements (we leave it in exponent form since it is astronomically large).
Worksheets — Subsets, Cardinality & Power Sets
Subsets and Element Membership
Focuses on understanding basic containment relationships ($\subseteq$ vs $\not\subseteq$) and checking symbols ($\in$ vs $\subseteq$).
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1
If $A = \{1, 2, \{3, 4\}\}$, identify whether the following statements are True or False:
(a) $2 \in A$ (b) $\{3, 4\} \subseteq A$ (c) $\{1, 2\} \subseteq A$ (d) $3 \in A$ -
2
List all the subsets of the set $A = \{a, 1\}$. How many subsets are proper subsets?
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3
Find the cardinality of the following sets:
(a) $X = \{1, 3, 5, 7, \dots, 99\}$ (b) $Y = \{x \mid x \text{ is an integer and } x^2 = 5\}$
Power Sets & Cardinality Equations
Testing subset counting formula $2^n$ and constructing power sets for nested and empty sets.
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4
Let $A = \{a, b, c\}$. Construct the power set $P(A)$. Check if $b \in P(A)$ and if $\{b\} \in P(A)$.
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5
Find the value of $n[P(P(\emptyset))]$ by evaluating step-by-step.
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6
Suggest an appropriate universal set $U$ for the following sets:
$A = \{x \mid x \text{ is a student in your school's football team}\}$,
$B = \{x \mid x \text{ is a student in your school's chess team}\}$.
Rigorous Cardinality Proofs & Nested Elements
Designed for higher order thinking, nested sets, and mathematical proofs on subset transitivity.
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7
Prove that if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$ (Transitivity of Subsets).
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8
Evaluate the cardinality of the set $M = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$. List all of its subsets.
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9
Let $A$ be a set with $k$ elements. If a new element is added to $A$ to form set $B$, explain mathematically why the number of subsets doubles.
Quick Revision Notes — Subsets, Power Sets & Cardinality
60-Second Summary
The core takeaways in under a minute
A set $A$ is a subset ($A \subseteq B$) of $B$ if every element in $A$ is also in $B$. It is a proper subset ($A \subsetneq B$) if $A \subseteq B$ and $A \neq B$. The empty set $\emptyset$ is a subset of every set. The cardinality $n(A)$ is the number of distinct elements in $A$. The power set $P(A)$ is the set of all subsets of $A$. If $n(A) = p$, then $n[P(A)] = 2^p$ subsets, and there are $2^p - 1$ proper subsets. The universal set $U$ acts as the bounding system for all sets in a given context.
Key Terminology & Symbols
Set relationship vocabulary
| Symbol / Term | Meaning | Example |
|---|---|---|
| $\subseteq$ | Is a subset of (elements are contained in) | $\{2, 3\} \subseteq \{1, 2, 3\}$ |
| $\subsetneq$ | Is a proper subset of (subset, and not equal to) | $\{2, 3\} \subsetneq \{1, 2, 3\}$ |
| $n(A)$ | Cardinality of set A (number of distinct elements) | $n(\{1, 1, 2, 3\}) = 3$ |
| $P(A)$ | Power set of A (set of all subsets) | $P(\{1\}) = \{\emptyset, \{1\}\}$ |
| $2^p$ | Number of subsets of a set with $p$ elements | A set with 3 elements has $2^3 = 8$ subsets. |
| $U$ | Universal Set (context bounds) | Real numbers ($\mathbb{R}$) for number sets. |
Common Mistakes to Avoid
Watch out for these classic exam traps!
- $\in$ relates an element to a set (e.g. $1 \in \{1\}$).
- $\subseteq$ relates a set to another set (e.g. $\{1\} \subseteq \{1\}$). Writing $1 \subseteq \{1\}$ is a syntax error!
If $A = \{1, \{2, 3\}\}$, then $2 \in A$ is FALSE. The element is $\{2, 3\}$, not $2$. Therefore, $\{2, 3\} \in A$ is true, and $\{\{2, 3\}\} \subseteq A$ is true.
The notations $\emptyset$ and $\{\}$ represent the same empty set. A set like $\{\emptyset, \{\}\}$ simplifies to $\{\emptyset\}$ which has cardinality 1, not 2.
Subsets, Cardinality & Power Sets Chapter Tests
Take a Practice Test
Select your testing level to practice subsets, proper subsets, cardinality calculations, power set subsets count, and universal sets. Each test is out of 24 Marks and contains 10 structured questions. Compare your responses with marking keys to self-grade descriptive parts.
Basic Test
Subset verification, element cardinality counts, and power set list of small sets.
Standard Test
Subset formulas, cardinality equations, rational set checks, and universal set definitions.
Advanced Test
Nested set subsets, subset transitivity proofs, power sets of empty sets, and advanced exponential equations.