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.
$$\Rightarrow 2^5$$