Subset

Subset

A set is a well-defined collection of numbers, alphabets, objects, or any items.

A subset is a part of the set.

If \(\begin{align} Set\:1 = \left \{ {A,B,C} \right \}\end{align}\) and \(\begin{align} Set\:2 = \left \{ {A,B,C,D,E,F} \right \}\end{align}\) we can say that \(\begin{align} Set \:1 \end{align}\) is a subset of \(\begin{align} Set \:2 \end{align}\) since all the elements in set 1 are present in set 2

Subset example - set 1 is a subset of set 2

In this lesson, you will learn about subsets, subset symbols, types of subsets, and their properties. You will also learn the differences between proper subset and improper subset. You will go through some solved examples and also be able to solve some interesting practice questions toward the end of the lesson. 

Lesson Plan

What Is a Subset?

A set is a collection of different objects.

Set A is a subset of set B if all the elements in set A are present in set B.

Here, A is called the subset, and B is called the superset. 

For example, let \(\begin{align} Set\: A \:=\: \{{All\: whole\: numbers} \}\end{align}\), \(\begin{align} Set\: B \:=\:\{ {All\: natural\: numbers}\} \end{align}\). We can say that set A is the super set of set B.

Super set and subset example - whole numbers is the super set and natural numbers is the subset


What Is the Symbol of a Subset?

  • The symbol for subset is '\(\begin{align} \subseteq\end{align}\)'
  • It is read as "is a subset of."

For example, if \(\begin{align} Set A = \left \{ {1,2,3} \right \}\end{align}\) and \(\begin{align} Set B = \left \{ {1,2,3,4,5,6} \right \}\end{align}\) we can say: 

\(\begin{align}A\sqsubseteq B\end{align}\) (read as A is a subset of B)


What Are the Types of Subsets?

There are two types of subsets. 

  • Proper subsets
  • Improper subsets

Proper Subset  

  • A subset is said to be a proper subset if at least one of the elements in set A is present in set B
  • It is denoted by the symbol "\(\begin{align} \subset \end{align}\)"  
  • The formula to calculate number of proper subsets is \(\begin{align} 2^n -1 \end{align}\) (where n is the number of elements in the set)

Example: \(\begin{align}  A = \left \{ {10,20} \right \}\end{align}\)

Total number of elements \(\begin{align}(n) = 2 \end{align}\) 

Number of proper subsets  = \(\begin{align}2^2-1 \end{align}\) 

                                        = \(\begin{align}4-1 \end{align}\)

                                        = \(\begin{align}3 \end{align}\)

Therefore, there are 3 elements in the proper subset which are \(\begin{align}\left \{ \left \{  \right \},\left \{ 10 \right \},\left \{ 20 \right \} \right \}\end{align}\)

Improper Subset

  • A subset that has every element of the other set is called an improper subset.
  • It is denoted by the symbol "\(\begin{align} \sqsubseteq  \end{align}\)"

Example: Set X = \(\begin{align}\left \{ 5,10,15 \right \}\end{align}\)

The subsets of set X are:

\(\left \{ {} \right \}\),\(\left \{ {5} \right \}\),\(\left \{ {10} \right \}\),\(\left \{ {15} \right \}\),\(\left \{ {5,10} \right \}\),\(\left \{ {5,15} \right \}\),\(\left \{ {10,15} \right \}\),\(\left \{ {5,10,15} \right \}\)

Here the subset {5,10,15} has all the elements of the original set and therefore we can say 

\(\begin{align}\left \{ {5,10,15} \right \} \sqsubseteq X \end{align}\)

 
Challenge your math skills
Challenging Questions
  1. Set A = List of all prime numbers from 1 to 10
    Set B = All the numbers from 1 to 10
    List out all the subsets and proper subsets.
  2. Set P = {'S', 'U', 'B', 'S', 'E', 'T'}. Find the number of subsets.

What is a Power Set?

  • A power set is a collection of all the subsets together along with the empty set and the original set.
  • It is denoted by the notation P(A) for a set A.
  • Given there are \(n\) number of elements in a set, then there will be exactly 2elements in the power set.

Example

Let us understand this with an example.

Set \(\begin{align}  X = \left \{ {2,4,6} \right \}\end{align}\)

Total number of elements in the set \(\begin{align}  X =3\end{align}\)

The power set of X is 

\( P( X )= \) \(\left \{ \right \}\),\(\left \{ 2 \right \}\),\(\left \{ 4 \right \}\),\(\left \{ 6 \right \}\),\(\left \{ 2,4 \right \}\),\(\left \{ 4,6 \right \}\),\(\left \{ 2,6 \right \}\),\(\left \{ 2,4,6\right \}\)

\(\begin{align} P( X )= 2^n\end{align}\) 

Substituting \(\begin{align} n\:=\:3 \end{align}\)

\(\begin{align} P( X )= 2^3\end{align}\)


What Are the Properties of Subsets?

  • Any given set is a subset of itself. If \(\begin{align} Set \:A = \left \{ {1,2,3} \right \}\end{align}\), we can say that set A is always a subset of set A, which means A \(\begin{align} \subset\end{align}\) A.
  • An empty set is a subset of every set.
  • If A is a subset of B, then we can say that all the elements in A are present in B.
  • The set which has all its elements in the other set is called a subset and the original set is called a superset.
  • There are \(\begin{align} 2^n \end{align}\) subsets and \(\begin{align} 2^n - 1 \end{align}\) proper subsets for a given set. 
 
important notes to remember
Important Notes
  1. An empty set is always a subset of any given set.
  2. The original set is a subset of its own.
  3. A proper subset has all the combinations of elements except the original set.
  4. The number of elements of a power set is \(\begin{align} 2^n\end{align}\) where 'n' is the number of elements in the set.

Solved Examples

Example 1

 

 

If \(\begin{align} X =\{ { a, b, c }\}\end{align}\), how many subsets and proper subsets will set X have? List all of them.

Solution

Number of subsets = \(\begin{align} 2^n\end{align}\)   

Number of subsets for set \(\begin{align} X = 2^3 = 8\end{align}\), where \(n\) is the number of elements of the set

The subsets of X are \(\begin{align}\left \{  \right \},\left \{ a\right \},\left \{ b \right \},\left \{ c \right \},\left \{ a,b \right \},\left \{ b,c \right \},\left \{ a,c\right \},\left \{ a,b,c \right \}\end{align}\)

Number of proper subsets = \(\begin{align}2^n - 1\end{align}\)

Number of proper subsets for set \(\begin{align} X = \end{align}\) \(\begin{align}2^3 - 1\end{align}\)

                                                     \(\begin{align} = 8 - 1 = 7\end{align}\)          

Subsets of X are \(\begin{align}\left \{  \right \},\left \{ a\right \},\left \{ b \right \},\left \{ c \right \},\left \{ a,b \right \},\left \{ b,c \right \},\left \{ a,c\right \},\left \{ a,b,c \right \}\end{align}\)

Proper subsets of X are \(\begin{align}\left \{ \right \},\left \{ a\right \},\left \{ b \right \},\left \{ c \right \},\left \{ a,b \right \},\left \{ b,c \right \},\left \{ a,c\right \}\end{align}\)

Example 2

 

 

Find the relationship between the sets A, B, and C.

\(\begin{align}  A = \left \{ {2,3,4} \right \}\end{align}\), \(\begin{align}  B = \left \{ {1,2,3,4,5} \right \}\end{align}\), \(\begin{align}  C = \left \{ {1,2,3,4,5} \right \}\end{align}\)

Solution

A is a subset of B since every element of set A is present in set B (A ⊆ B) 

A is a proper subset of B since every element in A is also in B and also set A is not equal to set B.

C is a subset of B, but we cannot say it is a proper subset, since set A is equal to set B (C ⊆ B) 

\(\begin{align}A\sqsubseteq B\end{align}\)and \(\begin{align}C\sqsubseteq B\end{align}\)

Example 3

 

 

Find the power sets of the set \(\begin{align}  P = \left \{ {3,5,7} \right \}\end{align}\).

Solution

There can be \(\begin{align} 2^n\end{align}\) power sets for a given set. 

\( P( X )= \) \(\left \{ \right \}\),\(\left \{ 3 \right \}\),\(\left \{ 5 \right \}\),\(\left \{ 7 \right \}\),\(\left \{ 3,5 \right \}\),\(\left \{ 5,7 \right \}\),\(\left \{ 3,7 \right \}\),\(\left \{ 3,5,7\right \}\)
Example 4

 

 

Set A = {1,3,5}

Set B = {1,3,5,7,9}

Find the relationship between the sets.

Relationship between sets A and B

Solution

We see that all the elements in set A are present in set B. 

\(\begin{align}A \sqsubseteq B\end{align}\)
Example 5

 

 

Set P = {a,c,d}

Set Q = {b,c,d,e,f}

Find the relationship between the sets.

Relationship between sets P and Q

Solution

We see that not all elements of set P are present in set Q.

Thus, we can clearly say that P is not a subset of Q.

\(\begin{align}P \nsubseteq Q\end{align}\)
Example 6

 

 

Set X = {1,2,3,4,5} 

Set Y = {3,4,2,1,5}

Find the relationship between the sets.

Relationship between sets - X and Y have the same set of elements but different order

Solution

We can see that the elements in set X and set Y are the same though they are not in the same order.

\(\begin{align}X \sqsubseteq Y\end{align}\) and \(\begin{align}Y \sqsubseteq X\end{align}\)

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of subsets. The math journey around subsets starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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At Cuemath, our team of math experts are dedicated to making learning fun for our favorite readers, the students!

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Be it problems, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions(FAQs)

1. Is an empty set a subset?

Yes, an empty set is a subset of every set since there are no elements in an empty set and it is a part of every set.

2. Is Phi a proper subset?

Yes, Phi is a proper subset. It is included every time when we write elements of a proper subset.

3. Is 0 an element of an empty set?

An empty set does not have any values. Hence, 0 is not an element of an empty set.

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