What Is The Subset Of

Subsets are a part of 1 of the mathematical concepts called Sets. A set is a collection of objects or elements, grouped in the curly braces, such as {a,b,c,d}. If a set A is a drove of even number and set B consists of {2,4,6}, and then B is said to be a subset of A, denoted by B⊆A and A is the superset of B. Learn Sets Subset And Superset to understand the difference.

The elements of sets could be anything such every bit a grouping of existent numbers, variables, constants, whole numbers, etc. It consists of a null set up likewise. Let us discuss subsets here with its types and examples.

Table of contents:

Definition
- Symbol
- All subsets
Types
Proper Subset
- - Proper Subset Symbol
  - Formula
- Subsets and Proper Subsets
Improper Subsets
- Power set up
Properties
Solved Examples
FAQs

What is a Subset in Maths?

Fix A is said to be a subset of Set B if all the elements of Set A are also present in Set B. In other words, gear up A is contained inside Set B.

Example: If set up A has {10, Y} and set B has {X, Y, Z}, then A is the subset of B because elements of A are also present in set B.

Subset Symbol

In fix theory, a subset is denoted past the symbol ⊆ and read as 'is a subset of'.

Using this symbol we can limited subsets every bit follows:

A ⊆ B; which ways Set A is a subset of Fix B.

Notation: A subset can be equal to the gear up. That is, a subset tin can comprise all the elements that are present in the set.

All Subsets of a Set

The subsets of whatever set consists of all possible sets including its elements and the null set. Let the states understand with the help of an example.

Case: Observe all the subsets of set A = {1,two,3,four}

Solution: Given, A = {ane,2,three,four}

Subsets =

{}

{one}, {2}, {three}, {iv},

{i,2}, {i,3}, {1,4}, {two,3},{2,four}, {three,4},

{1,2,3}, {ii,3,4}, {ane,3,iv}, {i,2,four}

{1,two,3,4}

Types of Subsets

Subsets are classified as

Proper Subset
Improper Subsets

A proper subset is one that contains a few elements of the original gear up whereas an improper subset, contains every element of the original set along with the null ready.

For example, if set A = {2, iv, vi}, then,

Number of subsets: {2}, {4}, {vi}, {2,4}, {4,6}, {2,6}, {2,4,half-dozen} and Φ or {}.

Proper Subsets: {}, {2}, {iv}, {6}, {2,4}, {4,six}, {2,6}

Improper Subset: {two,4,6}

There is no item formula to notice the subsets, instead, we have to list them all, to differentiate between proper and improper 1. The set theory symbols were developed by mathematicians to describe the collections of objects.

What are Proper Subsets?

Set A is considered to be a proper subset of Set B if Set B contains at least one chemical element that is non present in Set A.

Example: If prepare A has elements as {12, 24} and set B has elements as {12, 24, 36}, then set up A is the proper subset of B considering 36 is not present in the prepare A.

Proper Subset Symbol

A proper subset is denoted past ⊂ and is read as 'is a proper subset of'. Using this symbol, nosotros can express a proper subset for set A and set B as;

A ⊂ B

Proper Subset Formula

If we accept to option n number of elements from a set containing N number of elements, it tin can be washed in ^NC_nnumber of means.

Therefore, the number of possible subsets containing north number of elements from a set containing N number of elements is equal to ^NorthC_n.

How many subsets and proper subsets does a set take?

If a set has "n" elements, so the number of subset of the given set is two ⁿ and the number of proper subsets of the given subset is given by two ^{due north} -1.

Consider an case, If set up A has the elements, A = {a, b}, and so the proper subset of the given subset are { }, {a}, and {b}.

Here, the number of elements in the prepare is 2.

We know that the formula to summate the number of proper subsets is 2 ⁿ – one.

= 2 ² – one

= 4 – 1

= 3

Thus, the number of proper subset for the given set is 3 ({ }, {a}, {b}).

What is Improper Subset?

A subset which contains all the elements of the original set is called an improper subset. It is denoted by ⊆.

For example: Set up P ={2,iv,half dozen} And so, the subsets of P are;

{}, {2}, {iv}, {6}, {2,4}, {iv,half-dozen}, {ii,6} and {2,4,six}.

Where, {}, {two}, {4}, {six}, {2,four}, {four,6}, {two,6} are the proper subsets and {2,4,6} is the improper subsets. Therefore, nosotros can write {two,iv,6} ⊆ P.

Notation: The empty ready is an improper subset of itself (since it is equal to itself) but it is a proper subset of any other ready.

Power Set

The power set is said to be the collection of all the subsets. It is represented by P(A).

If A is set up having elements {a, b}. Then the power gear up of A will be;

P(A) = {∅, {a}, {b}, {a, b}}

To learn more in brief, click on the article link of power set.

Properties of Subsets

Some of the of import properties of subsets are:

Every set is considered as a subset of the given set itself. It means that 10 ⊂ 10 or Y ⊂ Y, etc
Nosotros tin say, an empty set is considered as a subset of every set.
X is a subset of Y. It means that X is contained in Y
If a set Ten is a subset of set Y, we can say that Y is a superset of X

Video Lesson on What are Sets

Likewise, read:

Sets For Course 11
Sets Subset And Superset
Union Of Sets
Universal Set

Subsets Example Issues

Example 1: How many number of subsets containing 3 elements tin be formed from the set?

S = { 1, ii, 3, 4, 5, half dozen, 7, viii, 9, 10 }

Solution: Number of elements in the set = ten

Number of elements in the subset = 3

Therefore, the number of possible subsets containing 3 elements = ^tenC_iii

\(\brainstorm{array}{l}=\frac{10!}{(10-three)!\times3!}\terminate{array} \)

\(\begin{array}{l}=\frac{x\times 9\times 8\times7!}{vii!\times3\times 2\times1}\end{array} \)

\(\begin{array}{l}=\frac{720}{6}=120\cease{array} \)

Therefore, the number of possible subsets containing three elements from the gear up S = { one, 2, 3, four, 5, 6, 7, eight, 9, 10 } is 120.

Case ii: Given any 2 existent-life examples on the subset.

Solution: We tin notice a multifariousness of examples of subsets in everyday life such as:

If we consider all the books in a library equally one fix, then books pertaining to Maths is a subset.
If all the items in a grocery shop class a set, then cereals class a subset.

Example iii: Find the number of subsets and the number of proper subsets for the given fix A = {five, 6, vii, viii}.

Solution:

Given: A = {5, 6, 7, 8}

The number of elements in the set is iv

We know that,

The formula to summate the number of subsets of a given set is 2 ^{due north}

= ii ^iv = 16

Number of subsets is 16

The formula to calculate the number of proper subsets of a given set is ii ^northward – 1

= two ^iv – 1

= 16 – ane = 15

The number of proper subsets is fifteen.

Often Asked Questions on Subsets

Define subset

In set theory, a prepare X is divers as a subset of the other set Y, if all the elements of ready 10 should be present in the set Y. This tin be symbolically represented by 10 ⊂ Y

What are the two classifications of subset?

The classifications of subsets are:
Proper subset
Improper subset

Define proper and improper subsets.

An improper subset is defined every bit a subset which contains all the elements present in the other subset. But in proper subsets, if X is a subset of Y, if and only if every element of set Ten should be present in set Y, but in that location is one or more elements of set Y is not present in set Ten.

Give an example of proper and improper subsets.

Proper subset:
X = {2, 5, 6} and Y = {2, 3, 5, 6}
Improper Subset:
X = {A, B, C, D} and Y = {A, B, C, D}

What is the formula to calculate the number of subsets and proper subset for any given ready?

If "north" is the number of elements of a given ready, and so the formulas to calculate the number of subsets and a proper subset is given by:
Number of subsets = twoⁿ
Number of proper subsets = 2ⁿ– 1

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