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Subsets and Power Sets – JEE Important Topic

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Last updated date: 18th Jul 2024
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What are Sets, Subsets and Power Sets?

In mathematics, a set refers to a bunch of numbers which are arranged in ordered pairs. A sub-set refers to a set of numbers that do not have all the possible elements of the parent set but the elements here are also a part of the main set of numbers. A power set refers to the set of elements that includes all possible subsets of the given parent set including a null set and the parent set itself. Let us take an example to further understand this concept. Let the set S = {x, y, z}. From the given parent set, the number of subsets that can be obtained are {x}, {y}, {z}, {x,y}, {y,z}, and {z,x}. All these sets are essentially a part of the main set S. The power set of the parent set S will have all these elements as well as the null set { }. Therefore, in this context the power set of the parent set S= ( {}, {x}, {y}, {z}, {x,y}, {y,z}, {z,x}, {x, y, z} ). 


Types of Subsets

A subset is defined as a set of elements that can have all, none or some of the elements of the parent set. A subset can be classified into either a proper subset or an improper subset. The symbol of the subset is represented by $\subset$.

Proper Subset Definition: Let us consider set A and set B. Set A will be considered a proper subset of set B if and only if set B has at least one element that is not present in set A. A proper subset is represented by the symbol $\subset$. The formula for the number of subsets is given by 2n.

Let us consider the following example of the proper subset.

Let set A contain the elements {5, 6}. And let set B contain the elements {5, 6, 7}. Since set B has one element extra or different from that of set A which is 7. Therefore, set A will be considered a proper subset of set B i.e., $A\subset B$

Improper Subset Definition: An improper subset is defined as that subset of a parent set that has all the elements of the main set. The symbol of an improper subset is $\subseteq$.  Let us consider the following example.

Let set A contain the elements {5,6,7}. Let set B have the elements {5,6,7}. Since set B contains all the elements of set A. Therefore, set B is an improper subset of set A. Therefore, $B \subseteq A$.

The formula for the number of subsets of a set possible from a set A with n elements is given by 2nand the number of proper sets that can be obtained is given by the formula 2n-1.


Theorems Related to Subsets

Three theorems must be kept in mind with regard to the concept of subsets and power sets. These theorems are as follows.

Theorem 1: Let A, B and C be the three sets taken into consideration. If $A \subseteq B$ and $B \subseteq C$ then, $A \subseteq C$.

Proof:

Given that A, B and C are the three sets.

We take the assumption that $A \subseteq B$ and $B \subseteq C$.

Let x be any arbitrary element of set A such that $x\in A$.

Since, $A \subseteq B$, therefore, $x\in B$. 

Again since $B \subseteq C$, therefore $x\in C$.

Since every element of A (x) is also an element of C, therefore, $A \subseteq C$.

Thus Proved.

Theorem 2: For any set A, we have $\{\}\subseteq A$ and $A\subseteq A$. In particular $\{\}\subseteq \{\}$

Proof:

As every element of A also appears in the same set A, therefore it is established that $A\subseteq A$. 

Let x be any arbitrary element such that it belongs to the universal U, $x\in U$.

Since { } is an empty set or a null set, therefore $x\in \{\}$always holds false. 

Therefore, $x\in \{\}\Rightarrow x\in A$.

Thus, $\{\}\subseteq A$ for any set A. Therefore, when A = { } we get that $\{\}\subseteq \{\}$

Hence proved.

Theorem 3: If A is a set containing n elements then the total number of subsets possible from the set A is 2n.

Proof:

When we select the number of elements in the subset of the parent set, there are two choices: either an element will be there or not. Since there are 2 distinct choices, therefore, for a set A containing n elements, the number of possible subsets will be 2n.

Hence proved.


Examples Related to Subsets 

Example 1: Set A = {a, b, c, d}. How many subsets and proper subsets can be obtained from the given set?

Solution:

The given set, set A = {a, b, c, d} 

Therefore the total number of elements in the given set A = 4 

Therefore the total number of subsets possible from this particular set = 24 = 16

Therefore the total number of proper subsets obtained from the given set = 24 - 1 = 16 - 1 = 15.


Example 2 : If set A= {a, b, c, d} and set B= {a, b}, how are the two sets related?

Solution:

Given set A= {a, b, c, d}

Given set B= {a,b}.

Comparing the two sets we get that the elements a and b are common to both sets A and B but the elements c and d are only a part of the set A. 

Since set B has two elements of the set A but not all, therefore set B is a proper subset of the set A which is represented by the expression $B\subset A$. 


Conclusion

A set A is subset of set B if and only if, the elements of set A are also present in set B. If set B has at least one more element that set A does not have, then set A is termed as a proper subset of set B. if all the elements of set B are present in set A, then set A is termed as an improper subset of set B. The formula of the number of subsets of a particular set having n elements is given by 2n. To define the power set of a set it is stated that when a set contains all the possible subsets of the parent set along with the main set itself and the null set, it is termed the power set.

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FAQs on Subsets and Power Sets – JEE Important Topic

1. What are the properties of subsets?

Following are the properties of the subsets:

  1. Every set is a subset of itself. A null set also known as a void or an empty set is a subset of all sets. 

  2. All the sets are a subset of the universal sets and if A is a subset of B, then by default B is the superset of A, and A is contained in B. 

  3. Every set is thought to be a subset of the given set. Its means that $A\subset A$ and $B\subset B$ etc.

  4. Set A is a subset of set B. It denotes that A is found in B.

2. What is the importance of set theory in real life?

The physical manifestation of the mathematical concept of set theory can be seen everywhere. For instance, a kitchen cabinet has separate places for plates, bowls and spoons; a shopping complex has specified areas for groceries and stationaries.

In both these cases,  the bowls, spoons and plates are all subsets of the kitchen while the shopping complex is the super set for the grocery section as well as the stationary section. Set theory is significant because it supplies the principles on which the rest of mathematics is based.