Subsets: Definition, Number of subsets of a set & Examples

Subsets: A set is a group of well-defined objects or elements generally written within a pair of curly braces, such as \(\left\{{a,b,c,d} \right\}.\) Subsets are considered a part of all the elements of the sets.

In this article, we shall focus on subsets by elaborating subsets, types of subsets, the number of subsets of a set, classification of subsets, and some solved examples and frequently asked questions.

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Definition of a Set

A set is defined as a collection of well-defined objects or elements, separated by commas, generally written within a pair of curly braces.

Example: The days in a week are well defined, such as Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday. These days can be written in the form of a set as \(A = \left\{{{\text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}}} \right\}{\text{.}}\)

Definition of a Subset

A subset is considered a set that contains part of the elements of a set or all the elements of a set. It is also written in the same way as a set.

Example: In the above example, if we may consider subsets in various ways. One of the ways is “collection of days of a week starting with the letter \(”T”.\) Thus, we have Tuesday and Thursday, which may be represented as \(B = \left\{{{\text{Tuesday, Thursday}}} \right\}.\)
In this case, we can say that the set \(B\) is a subset of the set \(A.\) We represent it as \(B \subset A.\)

Classification of Subsets

Subsets are mainly classified into two types:

1. Proper Subset
2. Improper Subsets

Proper Subset

Set \(B\) is considered to be a proper subset of set \(A\) if set \(B\) does not contain all the elements of set \(A.\) This means there has to be at least one element in set \(A,\) which is not present in set \(B.\)
Example: Let \(A = \left\{{1,2,3,4,5,6,7,8,9} \right\}\) and \(B = \left\{{1,2,3,4,5,6,7} \right\}.\)

In this case, we note that the elements \(8\) and \(9\) are not present in the set \(B,\) but thay are present in the set \(A.\) Hence, set \(B\) is a proper subset of \(A\) and, we write \(B \subset A.\)

Practice Exam Questions

Improper Subset

If we have two sets \(A\) and \(B,\) and if all the elements present in the set \(A\) are also present in the set \(B,\) we can still say that the set \(A\) is an improper subset of the set \(B\) and vice versa.
Example: Let \(A = \left\{{1,2,3,4,5,6,7,8,9} \right\}\) and \(\left\{{x:x\,{\text{is}}\,{\text{a}}\,{\text{natural}}\,{\text{number}}\,{\text{less}}\,{\text{than}}\,10} \right\}.\)

In this case, we note that both sets contain the same elements \(1,2,3,4,5,6,7,8,9.\) Hence, in this case, the set \(A\) is considered as an improper subset of the set \(B,\) and we write \(A \subseteq B.\) And, since the set \(B\) also contains all the elements of the set \(A,\) we say that the set \(B\) is considered as an improper subset of the set \(A\) and we write \(B \subseteq A\)

Empty Set

The empty set is a unique set having no elements. The number of elements of the set is \(0.\) An empty set is represented by a pair of the conventional curly brackets \(\left\{{}\right\}\) or \(\phi \)
Example: If we define a set \(A\) as “days of a week starting with the letter \(”Z”,\) then we observe that there is no day(s) in a week, starting with the letter \(”Z”.\) Hence, set \(A\) is an empty set, and we write \(A = \phi .\)

The empty set is also termed as null set or void set, or zero sets.

An empty set is a subset of every set.

Number of Subsets of a Set

If a set has \(”n”\) elements, then the total number of subsets that can be derived from the set is \({2^n}\) Including the empty set \(\phi .\)
These \({2^n}\) Subsets contain the original set itself (which is an improper set), from where the subsets are derived. Hence, these \({2^n}\) subsets that we obtain, the total number of proper subsets is given by \({2^n} – 1.\)
Example: If set \(A\) has the elements, \(A = \left\{{a,b} \right\},\) then the subsets that can be derived from out of the set \(A\) are \(\left\{{} \right\},\left\{ a \right\},\left\{ b \right\}\) and \(\left\{{a,b} \right\}.\) We note that the last subset \(\left\{{a,b} \right\}.\) is the set \(A\) itself. Hence, \(\left\{{a,b} \right\}.\) is an improper subset. And the number of proper subsets are \(3\) (three) only.

Now, the set \(A\) had \(2\) elements. Hence, the total number of subsets is given by \({2^2} = 4,\) which are given by \(\left\{{} \right\},\left\{ a \right\},\left\{ b \right\}\) and \(\left\{{a,b} \right\}\) and the total number of proper subsets is given by \({2^2} – 1 = 4 – 1 = 3,\) which are given by \(\left\{{} \right\},\left\{ a\right\}\) and \(\left\{ b \right\}.\)

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Finding all the Subsets of a Given Set

Let us take the set \(A:\left\{{1,2,3} \right\}.\)
Here, the number of elements in set \(A\) is \(3.\) So, the total number of subsets (including both proper and improper subsets) will be \({2^3} = 8.\)
The list of these \(8\) subsets are: \(\left\{{} \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{{1,2} \right\},\left\{{2,3} \right\},\left\{{1,3} \right\}\) and \(\left\{{1,2,3} \right\}.\)
If there are \(4\) objects in the set, then the subset will be \({2^4}.\)
Note: Power set: The set of all the subsets of a set is called a power set. In the above example, if the power set of the set \(A\) is \(P,\) then \(P = \left\{{\left\{{} \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{{1,2} \right\},\left\{{2,3} \right\},\left\{{1,3} \right\},\left\{{1,2,3} \right\}} \right\}\)

Solved Examples – Subsets

Q.1. How many numbers of elements are there in the power set of a set containing 5 elements?
Ans: The total number of elements in the power set of the set containing \(5\) elements is \({2^5} = 32.\)

Q.2. Give an example of proper and improper subsets.
Ans: Proper subset: \(X = \left\{{2,5,6} \right\}\) and \(Y = \left\{{2,3,5,6} \right\}.\) In this case, the set \(X\) is a proper subset of the set \(Y\) because the set \(Y\) has element \(3,\) which is not present in set \(X.\) Improper Subset:\(P = \left\{{A,B,C,D} \right\}\) and \(Q = \left\{{A,B,C,D} \right\}.\) In this case, the set \(P\) is an improper subset of the set \(Q\) and vice versa because the two sets have the same element.

Q.3. Find the number of subsets and the number of proper subsets for the given set. A={5, 6, 7, 8}.
Ans: Given, \(A = \left\{{5,6,7,8} \right\}\)
The number of elements in the set is \(4\)
We know that the formula to calculate the number of subsets of a given set is \({2^n}\) and that for the number of proper subsets is \({2^n} – 1\)
Hence, the number of subsets \({2^n}4 = 16.\)
And, the number of proper subsets of the given set is \({2^n} – 1 = {2^n} – 1 = 15\)

Q.4. Sushma has four different colour bracelets: black (b), gold (g), white (w), and silver (s). She is deciding on which ones to wear today. What are her choices? How many choices does Priya have?
Ans: Think of the bracelets she has to choose from as a set \(B = \left\{{b,w,g,s} \right\}.\) Then, her choices are every possible subset of set \(B.\) Here is an organized list of them:

Hence, Sushma has a total of \({2^4} = 16\) choices.

Q.5. If P = {a:a is an even number} and Q ={b: b is a natural number}, then figure out the subset here.
Ans: As per given data, \(P = \left\{{2,4,6,8,10, \ldots ,20, \ldots } \right\}\) and \(Q = \left\{{1,2,,3,4 \ldots 10 \ldots 20 \ldots .} \right\}\)
As we see that the set \(Q\) includes all the elements of set \(P.\) So, \(P\) is the subset of \(Q\) or \(P \subset Q.\)

Summary

In this article, we explained what a set and subset is with examples. We discussed types of subsets that are proper subsets and improper subsets. We explained how to find the number of subsets and the sum of subsets. And, last but not least, we have solved some examples for better understanding the concept of subsets.

Learn About Different Types of Sets

Frequently Asked Questions (FAQ) – Subsets

Q.1. What are subsets of a set?
Ans: A set \(X\) is a subset of another set \(Y\) if all elements of the set \(X\) are elements of the set \(Y.\) In other words, the set \(X\) is contained inside the set \(Y.\) The subset relationship is denoted as \(X \subset Y.\)

Q.2. How many types of subsets are there? How many subsets are there in math?        
Ans: Subsets are classified as
1. Proper Subset
2. Improper Subsets

Q.3. Is zero a subset of every set?
Ans: Yes, it is the only subset of every set. When zero is identified with the empty set, it will be a subset of every set.

Q.4 Define proper and improper subsets.
Ans: An improper subset is a subset that contains all the elements present in one more subset. But in proper subsets, if \(X\) is a subset of \(Y,\) if and only if every element of set \(X\) be present in set \(Y,\) but there is one or more than objects of set \(Y\) is not present in set \(X.\)

Q.5. Give any three real-life examples on the subsets.
Ans: We can find a variety of examples of subsets in everyday life, such as:
1. If we consider all the books on a library as one set, then books pertaining to Mathematics is a subset
2. If all the items in a grocery form a set, then cereals are subsets.
3. If we take food, it has many subsets like vegetables, fruits, green leafy vegetables etc.

We hope this detailed article on subsets helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!