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
$\left\{ \phi \right\},\{ 1\} ,\{ 2\} ,\{ 3\} ,\{ 1,2\} ,\{ 1,3\} ,\{ 2,3\} ,\left\{ {1,2,3} \right\}$
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\{ \phi \right\},\{ 1\} ,\{ 2\} ,\{ 3\} ,\{ 1,2\} ,\{ 1,3\} ,\{ 2,3\} $