Subsets and types of subsets

Relative to the set theory, 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 a particular segment to be focused, such as the genre of fiction, then this set refers to a part of the entire concept of a library. In the general overall categorization of books, if we look only into the fiction section of the library, then this will make up the proper subset of science fiction books because it contains some but not all of the books from that category.

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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.

Introduction to Subsets

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 \Leftrightarrow \forall x(x \in \dot{A} \Rightarrow x \in B)
$

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 c 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 ${ }^N \mathrm{C}_{\mathrm{n}}$.

_{Improper Subset
An improper subset is simply a subset that can be equal to the original set. By definition, every set is an improper subset of itself. Therefore, $A$ is an improper subset of $B$ if $A \subseteq B$.}

<sub>Intervals
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 the 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. $[a, b]=\{x \in \mathbb{R} \mid a \leq x \leq b\}$</sub>

<sub>1. Half-Open (or Half-Closed) Interval:
- Left Half-Open Interval [a,b): This includes all numbers between $a$ and $b$, including a but not $b$.</sub>

_{$
[a, b)=\{x \in \mathbb{R} \mid a \leq 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 \leq b\}
$}

<sub>2. Infinite Intervals: These extend indefinitely in one or both directions.
- Left Unbounded Interval $(-\infty, b)$ : This includes all numbers less than $\mathrm{b} .(-\infty, b)=\{x \in \mathbb{R} \mid x<b\}$
- Right Unbounded Interval $(a, \infty)$ : This includes all numbers greater than a. $(a, \infty)=\{x \in \mathbb{R} \mid x>a\}$
- Entire Real Line $(-\infty, \infty)$ : This includes all real numbers.</sub>

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

In set theory, collections are divided into graded structures comprised of subsets, proper subsets, improper subsets, and interval classes. It doesn’t matter whether it is bigger or not; a subset is just an assemblage that comes directly from a given set, for instance, the category of books within a library. A proper subset is a collection that contains a portion of the elements in the original set, for example, science fiction as a subcategory of fiction books while an improper subset entails all members of the set contained. Intervals also refer to continuous points making up a set, for instance, texts produced within a certain time.

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Solved Examples Based On the Subsests and Intervals:

Example 1: Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A x B, 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 $=\phi,(a e), \ldots \ldots(d f)$ and (ae, be) $\ldots \ldots \ldots=256$

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

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.

$
\begin{aligned}
& \mathrm{n}(\mathrm{A})=4, \mathrm{n}(\mathrm{B})=2 \\
& n(A \times B)=8
\end{aligned}
$

Number of subsets having at least 3 elements

$
\begin{aligned}
& ={ }^8 C_3+{ }^8 C_4+{ }^8 C_5+\ldots \ldots+{ }^8 C_8 \\
& =\left({ }^8 C_0+{ }^8 C_1+{ }^8 C_2+{ }^8 C_3+{ }^8 C_4+\ldots \ldots+{ }^8 C_8\right)-\left({ }^8 C_0+{ }^8 C_1+{ }^8 C_2\right) \\
& =2^8-\left(1+{ }^8 C_1+{ }^8 C_2\right) \\
& =219
\end{aligned}
$
Hence, the answer is 219.

Example 4: If a set $A$ has 8 elements, then the number of proper subsets of the $\operatorname{set} A$ is
Solution: $\square$
As we learned
Number of proper subsets $=2^8-1=256-1=255$
Hence, the answer is 255 .
Example 5: If a set has 32 subsets. How many elements does it have?
Solution: $\square$
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.

Frequently Asked Questions(FAQ)-

1. What is a subset?

Ans: A subset is a set in which all elements are also contained within another set. If $A$ and B are sets, then A is a subset of B (written as $A \subseteq B$ ) if every element of A is also an element of B.
2. What is a proper subset?

Ans: $A$ proper subset is a subset that is not equal to the original set. If $A$ and $B$ are sets, then A is a proper subset of B (written as $A \subset B$ ) if $A \subseteq B$ and $A \neq B$.
3. What is an improper subset?

Ans: An improper subset is a subset that is equal to the original set. In other words, every set is an improper subset of itself.

4. What are intervals in mathematics?

Ans: Intervals are a way to describe a set of real numbers between two endpoints. Intervals can be open, closed, or half-open. For example:
- A closed interval $[\mathrm{a}, \mathrm{b}]$ includes all numbers x such that $a \leq x \leq b$.
- An open interval $(a, b)$ includes all numbers $x$ such that $a<x<b$.
- A half-open interval [a, b) includes all numbers x such that $a \leq x<b$.
5. Can you provide numerical examples of subsets, proper subsets, improper subsets, and intervals?

Ans: Yes, here are some examples:
- Subsets: If $B=\{1,2,3,4\}$, then $A=\{2,4\}$ is a subset of $B$ because all elements of $A$ are in $B$.
- Proper Subsets: If $\mathrm{B}=\{1,2,3,4\}$, then $\mathrm{A}=\{1,2\}$ is a proper subset of B because $A \subseteq B$ and $A \neq B$.
- Intervals:
- The closed interval $[1,5]$ includes all real numbers from 1 to 5 , including 1 and 5 .