Subsets and types of subsets

The idea of subsets and the types of these subsets can be further elaborated through actual experiences. Suppose there is a library in which lots of books are available. If we consider the set of all mathematics books then this set refers to a part of the entire concept of a library. In this case set of mathematics books is the subset of the library of books. From this, we could say that the subset can be defined as the part of any set. In this article, let us look in detail about what is a subset in maths.

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This Story also Contains

1. Subsets
2. Types of Subset
3. Power Set of a Set
4. Intervals as Subset of $R$
5. Properties of Subsets
6. Subsets Examples

In this article, we will cover the concept of subsets, proper subsets, Improper subsets, and Intervals. This concept falls under the broader category of sets relation and function, a crucial chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of eight questions have been asked on this concept, including one in 2013, one in 2015, one in 2018, two in 2019, one in 2020, one in 2021, and one in 2023.

Subsets

They are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters $A,B,C,S,U,V$. Now, let us define subsets.

How do we define subset?

In set theory, a subset is a set whose elements are all contained within another set. If $A$ and $B$ are sets, $A$ is a subset of $B$ (denoted as $A\subseteq B$ ) if every element of $A$ is also an element of $B$. This concept can be formally expressed as:

$A\subseteq B\iff \mathrm{\forall }x(x\in \dot{A}\Rightarrow x\in B)$

Subset Symbol

In set theory, a subset is shown by the symbol ⊆ and read as ‘is a subset of’.

W ⊆ C; which means Set W is a subset of Set C.

All Subsets of a Set

We understand this concept with the help of example below:

Example: Find all the subsets of set R {a,b,c}

Solution: Given, R = {a,b,c}

Subsets are as follows:

{}

{a}, {b}, {c}

{a,b}, {b,c}, {a,c}

{a,b,c}

Types of Subset

Classifications of subset can be made as follows:

Proper Subset

A proper subset is a subset that is not equal to the set it is contained within. In other words, $A$ is a proper subset of $B$ (denoted as $A\subset B$ ).

Proper Subset Symbol

A proper subset is denoted by the symbol $\subset$ and is read as 'is a proper subset of'. Using this symbol, we can express a proper subset for set $A$ and set $B$ as $A\subset B$.

Proper Subset Formula

If we have to pick $n$ number of elements from a set containing N number of elements, it can be done in ${}^N{C}_n$ number of ways.

Therefore, the number of possible subsets containing $n$ number of elements from a set containing $N$ number of elements is equal to ${}^N{\mathrm{C}}_{\mathrm{n}}$.

How many subsets and proper subsets does a set have?

We consider a set has “n” elements, then number of subset = 2ⁿ and number of proper subsets = 2ⁿ-1.

For example, If set X has the elements, X = {1, 2}, then the proper subset of the given subset are { }, {1}, and {2}.The number of elements in the set is 2.

Number of proper subsets = 2ⁿ – 1.

= 2² – 1

= 4 – 1

= 3

Hence, number of proper subset for the given set is 3 ({ }, {1}, {2}).

_{Improper Subset}

An improper subsetis simply a subset that can be equal to the original set. By definiton, every set is an improper subset of itself.

Let's see some subset example,

1. Let $A=\{a,s,c\}$. The subsets of $A$ are $\{\},\{a\},\{s\},\{c\},\{a,s\},\{a,c\},\{s,c\},\{a,s,c\}$. Here, $\{a,s,c\}$ is the improper subset of $A$ while all others are proper subsets.

2. Let $B=\{2,3,4\}$. The possible subsets of $B$ are $\{\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\},\{2,3,4\}$. Here, $\{2,3,4\}$ is the improper subset of $A$ while all others are proper subsets.

Power Set of a Set

The power set of a set is defined as a set of all the subsets (along with the empty set and the original set). The power set of a set Y is denoted by P(Y). If Y has 'n' elements then P(Y) has 2ⁿ elements. For example,

• If E = {x, p}, then P(E) = { { }, {x}, {p}, {x, p} }
  Total number of elements = 4 .
• If A = {a, b, c}, then P(A) = { { }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c} }
  Observe that A has 3 elements and P(A) has 2³ = 8 elements.

Intervals as Subset of $R$

In mathematics, particularly in the context of real numbers, an interval is a set of numbers that lie between two specific numbers, known as the endpoints of that interval. There are several types of intervals:

1. Open Interval $(a,b)$ : This includes all numbers greater than $a$ and less than $b$, but not including $a$ and $b$.

$(a,b)=\{x\in \mathbb{R}\mid a<x<b\}$

2. Closed Interval $[a,b]$: This includes all numbers between $a$ and $b$, including $a$ and $b$.

$[a,b]=\{x\in \mathbb{R}\mid a\le x\le b\}$

3. Half-Open (or Half-Closed) Interval:

- Left Half-Open Interval $[a,b)$: This includes all numbers between $a$ and $b$ including $a$ but not $b$

$[a,b)=\{x\in \mathbb{R}\mid a\le x<b\}$

- Right Half-Open Interval $(a,b]$: This includes all numbers between $a$ and $b$ including $b$ but not $a$

$(a,b]=\{x\in \mathbb{R}\mid a<x\le b\}$

4. Infinite Intervals: These extend indefinitely in one or both directions.
- Left Unbounded Interval $(-\mathrm{\infty },b)$ : This includes all numbers less than $b$.
$(-\mathrm{\infty },b)=\{x\in \mathbb{R}\mid x<b\}$

- Right Unbounded Interval $(a,\mathrm{\infty })$: This includes all numbers greater than $a$.

$(a,\mathrm{\infty })=\{x\in \mathbb{R}\mid x>a\}$

- Entire Real Line $(-\mathrm{\infty },\mathrm{\infty })$ : This includes all real numbers.

$(-\mathrm{\infty },\mathrm{\infty })=\mathbb{R}$

Properties of Subsets

The properties of subset include the following points:

• Every set is always a subset of itself.
• Empty set is always a subset of every set.
• The set of all possible subsets of a set is known as the power set.
• The number of elements(cardinality) in a power set of a set containing $n$ elements is $2^n$.

Subsets Examples

Example 1: Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $AXB$, each having at least three elements is:

Solution:

As we learned in

SUBSETS -

$A$ set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$.
wherein
It is represented by $\subset$. eg. $A\subset B$ if $A=\{2,4\}$ and $B=\{1,2,3,4,5\}$
Let $A$ having elements $\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}$ and $B$ having $\{\mathrm{e},\mathrm{f}\}$
Then $A\times B$ having 8 elements and no. of subsets $=2^8=256$
No. of subsets $=\varphi ,(ae),\dots \dots (df)$ and (ae, be) $\dots \dots \dots =256$

Now $8$ subsets have only one element

$\{ae\},\{af\},\{be\},.......\{df\}$

Similarly

No of the sets having two elements

$\{ae,af\},\{ae,be\},.....\{ae,df\}=7$ elements

$\{af,be\},\{af,bf\}..........=6$ elements

$7+6+5+4+3+2+1=28$ (having two elements)

and a subset having a single element $\varphi$

$28+8+1=37$

At least three elements $=256-37=219$

Hence, the answer is $219.$

Example 2: If $A=\{1,2,3,5,7\}$ and $B=\{3,5\}$, then
1) $A\subset B$
2) $B\not\subset A$
3) $A=B$
4) $B\subset A$

Solution
All elements of $B$ are present in $A$ , thus $B\subset A$.
Hence, the answer is the option 4.

Example 3: If a set $A$ has $8$ elements, then the number of proper subsets of the $\mathrm{set}A$ is
Solution:
As we learned
Number of proper subsets $=2^8-1=256-1=255$
Hence, the answer is $255$.

Example 4: If a set has $256$ subsets. How many elements does it have?
1) $5$
2) $7$
3) $8$
4) $9$

Solution

$\begin{array}{rl}& 2^n=256\\ & \Rightarrow n=8\end{array}$

Hence, the answer is the option 3.

Example 5: If a set has $32$ subsets. How many elements does it have?
Solution:
As we know, if a set has $n$ elements, it will have $2^n$ subsets.
Thus $2^n=32$$\Rightarrow n=5$

Hence, the answer is $5$.

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