Subsets and Power Sets – JEE Important Topic

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.