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

Hint: 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.
\[ \Rightarrow {2^5}\]