Question

If A = \left\\{ {1,2,\left\\{ {3,4} \right\\}} \right\\}, then the number of non-empty proper subsets of $A$ is
A.16
B.15
C.3
D.6

Answer 1

Here, we are required to find the number of proper subsets of a given set $A$. We will use the formula of non-empty proper subsets to answer this question.
Formula Used:
We will use the formula for non-empty proper subsets$= {2^n} - 2$

Complete step-by-step answer:
If we are given two sets P and Q, then P is a subset of Q if all elements in P are there in Q but there should be no element which is there is P but is not present in Q.
Also, P would be a proper subset of Q if and only if it is not equal to Q i.e. there must be at least one element that is not there in the proper subset of Q, i.e. P.
Hence, a subset can have all the elements of a given set but a proper subset should not be equal to the given set.
Now, we are given a set $A$ such that:
A = \left\\{ {1,2,\left\\{ {3,4} \right\\}} \right\\}
Now, as we can see, there are three elements in set $A$, \left\\{ 1 \right\\},\left\\{ 2 \right\\},\left\\{ {3,4} \right\\}.
We have taken it as 3 elements because 3 and 4 are combined as one single element.
The formula to find the number of subsets of a given set $= {2^n}$
Now, we are required to find the number of non-empty proper subsets of $A$
Hence, the empty subset and the subset containing all the three elements would not be considered.
Therefore, the number of non-empty proper subsets of $A = {2^n} - 2$
Since, there are 3 elements in $A$, hence substituting,$n = 3$
Therefore, the number of non-empty proper subsets of $A$ $= {2^3} - 2$
Applying the exponent on the term, we get
$\Rightarrow$ Number of non-empty proper subsets of $A$ $= 8 - 2 = 6$
Hence, option D is the correct option.
There are 6 non-empty proper subsets of $A$.

Note: In this question, we can make mistakes such as; taking the number of elements as 4 and then forgetting to subtract 2 from the formula of subsets. And hence, making our answer completely wrong. Here, we are asked to find non-empty proper subsets of $A$, so we can make a mistake by finding subsets instead of non-empty proper subsets .