How To Find Proper Subsets

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

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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 office of all the elements of the sets.

In this commodity, we shall focus on subsets past elaborating subsets, types of subsets, the number of subsets of a fix, classification of subsets, and some solved examples and oftentimes asked questions.

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Learn All the Concepts on Sets

Definition of a Prepare

A ready is defined equally a collection of well-defined objects or elements, separated by commas, mostly written within a pair of curly braces.

Example: The days in a week are well divers, such equally Sunday, Mon, Tuesday, Midweek, Thursday, Fri and Sabbatum. These days tin be written in the form of a set as \(A = \left\{{{\text{Lord's day, Monday, Tuesday, Wednesday, Thursday, Friday, Sat}}} \right\}{\text{.}}\)

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Definition of a Subset

A subset is considered a gear up that contains role of the elements of a set or all the elements of a set. Information technology is also written in the aforementioned way equally a set.

Example: In the above example, if we may consider subsets in diverse means. Ane of the means is "collection of days of a week starting with the letter \("T".\) Thus, we take Tuesday and Th, which may be represented as \(B = \left\{{{\text{Tuesday, Th}}} \right\}.\)
In this case, we can say that the set \(B\) is a subset of the set \(A.\) Nosotros correspond information technology as \(B \subset A.\)

Classification of Subsets

Subsets are mainly classified into 2 types:

Proper Subset
Improper Subsets

Proper Subset

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

In this instance, we notation that the elements \(8\) and \(nine\) are not present in the set up \(B,\) merely thay are present in the gear up \(A.\) Hence, set \(B\) is a proper subset of \(A\) and, we write \(B \subset A.\)

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Improper Subset

If nosotros take ii sets \(A\) and \(B,\) and if all the elements present in the fix \(A\) are besides present in the set \(B,\) we tin still say that the gear up \(A\) is an improper subset of the set \(B\) and vice versa.
Case: Let \(A = \left\{{1,2,3,iv,5,6,7,viii,9} \right\}\) and \(\left\{{10:x\,{\text{is}}\,{\text{a}}\,{\text{natural}}\,{\text{number}}\,{\text{less}}\,{\text{than}}\,10} \right\}.\)

In this case, we annotation that both sets comprise the same elements \(1,2,three,iv,5,6,7,8,9.\) Hence, in this instance, the set \(A\) is considered as an improper subset of the ready \(B,\) 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 every bit an improper subset of the set \(A\) and nosotros write \(B \subseteq A\)

Empty Set

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

The empty fix is as well termed every bit nix fix or void fix, or zero sets.

An empty set is a subset of every set.

Number of Subsets of a Gear up

If a set has \("northward"\) elements, so the total number of subsets that can be derived from the set is \({2^n}\) Including the empty ready \(\phi .\)
These \({ii^n}\) Subsets contain the original prepare itself (which is an improper prepare), from where the subsets are derived. Hence, these \({2^north}\) subsets that we obtain, the total number of proper subsets is given by \({2^due north} – 1.\)
Example: If set \(A\) has the elements, \(A = \left\{{a,b} \correct\},\) then the subsets that can be derived from out of the set \(A\) are \(\left\{{} \right\},\left\{ a \right\},\left\{ b \correct\}\) and \(\left\{{a,b} \right\}.\) Nosotros note that the last subset \(\left\{{a,b} \correct\}.\) is the prepare \(A\) itself. Hence, \(\left\{{a,b} \right\}.\) is an improper subset. And the number of proper subsets are \(iii\) (three) but.

Now, the set \(A\) had \(two\) elements. Hence, the total number of subsets is given by \({ii^2} = 4,\) which are given by \(\left\{{} \right\},\left\{ a \right\},\left\{ b \correct\}\) and \(\left\{{a,b} \right\}\) and the full 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 \(A:\left\{{one,2,3} \right\}.\)
Hither, the number of elements in prepare \(A\) is \(3.\) So, the total number of subsets (including both proper and improper subsets) volition be \({2^3} = 8.\)
The listing of these \(8\) subsets are: \(\left\{{} \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ iii \correct\},\left\{{i,two} \correct\},\left\{{2,iii} \right\},\left\{{1,3} \right\}\) and \(\left\{{ane,two,3} \right\}.\)
If in that location are \(4\) objects in the set, then the subset volition be \({2^four}.\)
Note: Power prepare: The set up of all the subsets of a set is called a ability set. In the above example, if the power set of the prepare \(A\) is \(P,\) then \(P = \left\{{\left\{{} \right\},\left\{ ane \right\},\left\{ two \right\},\left\{ 3 \right\},\left\{{1,2} \right\},\left\{{2,3} \right\},\left\{{1,3} \right\},\left\{{1,two,3} \right\}} \right\}\)

Solved Examples – Subsets

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

Q.ii. Give an example of proper and improper subsets.
Ans: Proper subset: \(X = \left\{{ii,5,half-dozen} \right\}\) and \(Y = \left\{{2,3,5,6} \right\}.\) In this instance, the set \(X\) is a proper subset of the prepare \(Y\) considering the set \(Y\) has element \(3,\) which is non 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 ready \(Q\) and vice versa considering 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, half-dozen, 7, 8}.
A ns: Given, \(A = \left\{{five,6,7,viii} \right\}\)
The number of elements in the ready is \(iv\)
We know that the formula to calculate the number of subsets of a given ready is \({2^n}\) and that for the number of proper subsets is \({two^n} – 1\)
Hence, the number of subsets \({2^north}4 = 16.\)
And, the number of proper subsets of the given set is \({2^n} – 1 = {2^due north} – 1 = 15\)

Q.4. Sushma has four different color bracelets: black (b), gold (1000), white (west), and silver (s). She is deciding on which ones to article of clothing today. What are her choices? How many choices does Priya have?
Ans: Think of the bracelets she has to choose from as a set \(B = \left\{{b,westward,g,s} \right\}.\) Then, her choices are every possible subset of set \(B.\) Here is an organized list of them:

\(0\)chemical element subsets	\(1\)-chemical element subsets	\(2\)-element subsets	\(3\)-chemical element subsets	\(4\)-chemical element subsets
\(\left\{{} \right\}\) She decides not to article of clothing a bracelet	\(\left\{ b \correct\}\) \(\left\{ w \right\}\) \(\left\{ grand \right\}\) \(\left\{ southward \right\}\)	\(\left\{{b,w} \right\}\) \(\left\{{b,g} \right\}\) \(\left\{{b,s} \right\}\) \(\left\{{w,g} \right\}\) \(\left\{{w,south} \right\}\) \(\left\{{chiliad,s} \right\}\)	\(\left\{{b,west,g} \right\}\) \(\left\{{b,w,due south} \right\}\) \(\left\{{b,thousand,s} \right\}\) \(\left\{{due west,g,south} \right\}\)	\(\left\{{b,w,g,due south} \right\}\)

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

Q.5. If P = {a:a is an even number} and Q ={b: b is a natural number}, then effigy out the subset here.
Ans: As per given data, \(P = \left\{{two,four,half dozen,eight,10, \ldots ,xx, \ldots } \right\}\) and \(Q = \left\{{1,ii,,3,4 \ldots x \ldots 20 \ldots .} \right\}\)
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. Nosotros discussed types of subsets that are proper subsets and improper subsets. We explained how to observe the number of subsets and the sum of subsets. And, final but not least, we have solved some examples for better understanding the concept of subsets.

Learn About Dissimilar Types of Sets

Frequently Asked Questions (FAQ) – Subsets

Q.1. What are subsets of a gear up?
Ans: A gear up \(X\) is a subset of some other prepare \(Y\) if all elements of the set up \(X\) are elements of the prepare \(Y.\) In other words, the set \(X\) is independent 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 equally
1. Proper Subset
2. Improper Subsets

Q.iii. Is cipher a subset of every prepare?
Ans: Yep, 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.four 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 but if every element of set \(X\) be present in prepare \(Y,\) but there is one or more than objects of set \(Y\) is not present in gear up \(X.\)

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

We promise this detailed commodity on subsets helped you in your studies. If y'all take any doubts, queries or suggestions regarding this article, feel free to ask us in the comment department and nosotros will be more than happy to assist yous. Happy learning!