Question

The number of proper subsets of the set $\left\{1,2,3\right\}$is
A.$8$
B.$7$
C.$6$
D.$5$

Answer 1

Hint: In this question, we should first know the definition of sets and subsets. A set is a collection of objects or elements grouped in curly brackets. As for example:
$\left\{a,b,c,d\right\}$
While we should know that if a set A is a collection of even numbers and Set B consists of
$\left\{2,4,6\right\}$ , then we can see that Set B consists of elements that are part of even numbers. So we can say that set B is the subset of Set A .
Proper Subset: If we have set B, then set B is considered to be a proper subset of set A if all the elements of B are in A but A contains at least one element that is not in B.
 It is denoted by the symbol
$\subset$ .
As for example, we have set A :
 $A=\left\{1,3,5\right\}$ then we can say that set
$B=\left\{1,5\right\}$ is a proper subset of A .
The formula to calculate the number of proper subsets is
$2^n-1$ , where $n$ is the number of elements in the set .
So we will use this formula to solve the above question.

Complete step-by-step answer:
In this question we have, set
$\left\{1,2,3\right\}$.
We should note that in every set , there is one improper set that is the set itself.
So the formula that we will use to calculate the total number of subsets is
$2^n-1$ , where $n$ is the number of elements in the set .
So we can see that there are three elements in the given set i.e.
$n=3$
We can put this in the formula and we have:
$2^3-1$
By breaking the parts we have
$2\times 2\times 2-1$
It gives us the number of improper sets:
$8-1=7$
Hence the correct option is (B) $7$.
So, the correct answer is “Option B”.

Note: here is an alternative method to solve this question, we can write all the possible proper subsets of the given set
$\left\{1,2,3\right\}$and then we can check them.
We should note that if we have to find the number of subset, then the formula is
$2^n$ , where $n$ is the number of elements of the set.
Here we have three elements in the set, so we have
$n=3$ .
So the number of subsets are
$2^3=2\times 2\times 2=8$ subsets.
We can also write all the subsets. We should keep in mind that the empty set (or null) and the set itself is also a subset.
So the subsets are
$\left\{\varphi \right\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\left\{1,2,3\right\}$
Now we can write the proper subsets from the above.
We know that the empty set is a proper set of every set but the set is not a proper set of itself.
So the proper subsets are:
$\left\{\varphi \right\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\}$