Question

If $X = \\{ 1,2,3,.....,10\\}$ and $A = \\{ 1,2,3,4,5\\}$. Then, the number of subsets $B$ of $X$ such that $A - B = \\{ 4\\}$ is
A) ${2^5}$
B) ${2^4}$
C) ${2^5} - 1$
D) $1$
E) ${2^4} - 1$

Answer 1

We need to find the elements that could be in the set $B \subseteq X$ such that it satisfies the condition $A - B = \\{ 4\\}$.
Also we have to find the number of elements in it.
After that we can use the formula and we will get the required answer.

Formula used: Number of subsets of a set is given by ${2^n}$ where $n$ is the number of elements of the given set.

Complete step-by-step answer:
Given that $A - B = \\{ 4\\}$ implies that $\\{ 4\\} \in A$ and $\\{ 4\\} \notin B$.
Now we can take $\Rightarrow A - B = \\{ 4\\}$
Let us change the terms and we get,
$\Rightarrow B = A - \\{ 4\\}$
Since, it is stated as the question that $A = \\{ 1,2,3,4,5\\}$ ,
Now we can find $B$ value by removing $\\{ 4\\}$ from the set $A$.
Here we can write it as,
$\Rightarrow B = \\{ 1,2,3,4,5\\} - \\{ 4\\}$
After removing $\\{ 4\\}$ and we get,
$\Rightarrow B = \\{ 1,2,3,5\\} \to (1)$
Also, we need to find the number of subsets $B$ of $X$.
That is the number of subsets of $B \subseteq X$.
Now, we need to find the elements of $B$ that are also in $X$ and not in $A$ .
From (1) we have $B = \\{ 1,2,3,5\\}$ and from the question we have $X = \\{ 1,2,3,.....,10\\}$,
By taking $X - \\{ 4\\} - \\{ 1,2,3,5\\}$ should give the elements that are both in $B$ as well as $X$.
We consider $X - \\{ 4\\} - \\{ 1,2,3,5\\}$ because from the question we know that $\\{ 4\\} \notin B$ and $\\{ 1,2,3,5\\} \in A \cap B$ then these elements will not be in$B \subseteq X$.
By evaluating $X - \\{ 4\\} - \\{ 1,2,3,5\\}$ as
$\Rightarrow \\{ 1,2,3,4,5,6,7,8,9,10\\} - \\{ 4\\} - \\{ 1,2,3,5\\}$
Here we did not write the same term and we get the remaining,
$\Rightarrow \\{ 6,7,8,9,10\\}$
Then $B \subseteq X$ has 5 elements which are $\\{ 6,7,8,9,10\\}$. Which means that $\\{ 6,7,8,9,10\\}$ are the only numbers in $X$ which are also in$B$.
As we have to use the formula, we can now use ${2^n}$ to find the number subsets of $B \subseteq X$ and $n = 5$.
Hence the number of subsets $B$ of $X$ will be ${2^5}$.

Option A will be the correct answer for this question.

Note: We can also solve this problem using an alternative method as follows,
From the question $A - B = \\{ 4\\}$ implies that $\\{ 4\\} \in A$ and $\\{ 4\\} \notin B$ as we have already discussed.
Now, Number of elements in set X will be 10. That is, the number of subsets of X will be${2^{10}}$.
Since we have to find the number of subsets of $B \subseteq X$, We need to find the elements both in $X$ as well as $B$. Then,$\\{ 4\\}$ cannot be in $B$ of $X$ .
Number of subsets of $X$ that do not contain $\\{ 4\\}$ will be ${2^{10}} - {2^1} = {2^9}$
Also, we have seen that $\\{ 1,2,3,5\\} \in B$. Hence number of subsets of $X$ not containing $\\{ 4\\}$ and not containing $\\{ 1,2,3,5\\}$ will be ${2^{10}} - {2^1} - {2^4}$.
$\Rightarrow {2^{10}} - {2^1} - {2^4}$
Take $- 2$ as common and we can add the power we get,
$\Rightarrow {2^{10}} - {2^{(4 + 1)}}$
Since the powers with a common base can be added.
$\Rightarrow {2^{10}} - {2^5}$
Again, powers with a common base can be subtracted.
$\Rightarrow {2^5}$