Question

The total number of subsets of the set \[\left\{ {1,3,5,7} \right\}\] is
A.8
B.10
C.12
D.16

Answer 1

Hint: Number of subsets of a set is given by the formula \[ = {2^n}\] , where \[n\] is the number of elements in the set. In this question we are given a set, so first we will find the number of subset in the set and then by using the formula \[{2^n}\] we will determine the number of the subset that the given set can have.

Complete step-by-step answer:
Given the set \[S = \left\{ {1,3,5,7} \right\}\]
Now from the given set we can see there are 4 numbers of elements are there hence we can write \[n = 4\]
Now as we know the number of subset in the set can be determined by the formula \[{2^n}\] , hence by substituting the values we can write \[{2^n}\]
 \[
  Subset = {2^n} \\
   = {2^4} \\
   = 16 \;
 \]
Hence we can say the total number of subsets of the set \[\left\{ {1,3,5,7} \right\}\] is \[ = 16\]
To check
There are total of 16 subsets so now we will enlist them to verify
Write all the single element s as each element of a set is also its subset, we get \[\left\{ 1 \right\},\left\{ 3 \right\},\left\{ 5 \right\},\left\{ 7 \right\}\]
Now enlist subset with two elements \[\left\{ {1,3} \right\},\left\{ {1,5} \right\},\left\{ {1,7} \right\},\left\{ {3,5} \right\},\left\{ {3,7} \right\},\left\{ {5,7} \right\}\]
Now enlist subset with three elements \[\left\{ {1,3,5} \right\},\left\{ {1,3,7} \right\},\left\{ {3,5,7} \right\},\left\{ {1,5,7} \right\}\]
Now since there are total of 4 element s so this will be counted as one subset \[\left\{ {1,3,5,7} \right\}\]
Now we will list \[\left\{ \phi \right\}\] since an empty set is also a subset.
Hence we can say the number of subsets of the set \[\left\{ {1,3,5,7} \right\}\] is \[ = 16\]
So, the correct answer is “16”.

Note: The position of element in a set does not matter, if the element of a set is represented as \[\left\{ {a,b} \right\}\] or \[\left\{ {b,a} \right\}\] then the set is same. A subset of a set is the collection of all elements that are part of another set. We can see if the element of set A is \[\left\{ {p,q,r} \right\}\] and of set B is \[\left\{ {p,q} \right\}\] then set B is the subset of set A.