Question

If A, B, C are the three sets, \[A-(B\cap C)\] is equal to
(1) \[(A-B)\cup (A-C)\]
(2) \[(A-B)\cap (A-C)\]
(3) \[(A-B)\cup C\]
(4) \[(A-B)\cap C\]

Answer 1

Hint: If two sets are to be the subset of each other then both the sets must be equal. There are two ways of representing sets. The first one is in roaster form and the second one is in set builder form. The two sets A and B are said to be equal if every element of set A is in set B and vice versa.

Complete step-by-step answer:
Sets in mathematics are known as the well-defined collection of objects and the objects can be their members, elements, or points.
A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set. The order of a set defines the number of elements a set is having. It describes the size of a set.
There are various kinds of operations that we can perform on the sets and these are- the intersection of the sets and the union of the sets.
The operations of the sets are carried out when two or more sets combine to form a single set under some of the given conditions. These are –
1) Union of sets
2) Intersection of sets
3) A complement of a set
4) Cartesian product of set
5) Set difference
If we have to find the union of the set then all the elements will be the elements of either set A or in set B or can be in both the sets. If we have to find the intersection of the set then it will represent the elements that are common in set A and set B.
In the above question, it is given that A, B, and C are three sets and we have to find the value of \[A-(B\cap C)\] which is as follows.
We know that,
Any two sets are equal if and only if both are the subsets of each other, that is,
\[P\subset Q\] and \[Q\subset P\]
Now let us consider that
\[x\in A-(B\cap C)\]
\[\begin{align}
  & \Rightarrow x\in A,x\notin (B\cap C) \\
 & \Rightarrow x\in A,(x\notin B\text{ }or\text{ }x\notin C) \\
 & \Rightarrow (x\notin A,x\notin B)\text{ or (}x\in A,\text{ }x\notin C\text{)} \\
 & \Rightarrow x\in (A-B)\text{ }\cup \text{ }x\in (A-C) \\
 & \Rightarrow x\in (A-B)\text{ }\cup \text{ }(A-C) \\
\end{align}\]
So, \[A-(B\cap C)\subset (A-B)\cup (A-C)\]
Again, let us consider that
\[x\in (A-B)\text{ }\cup \text{ }(A-C)\]
\[\begin{align}
  & \Rightarrow x\in (A-B)\cup (A-C) \\
 & \Rightarrow x\in (A-B)\text{ or }x\in (A-C) \\
 & \Rightarrow (x\in A,x\notin B)\text{ or (}x\in A,x\notin C\text{)} \\
 & \Rightarrow x\in A,(x\notin B\text{ or }x\notin C) \\
 & \Rightarrow x\in A,x\notin (B\cap C) \\
 & \Rightarrow x\in A-(B\cap C) \\
\end{align}\]
So, \[A-(B\cap C)\subset (A-B)\cup (A-C)\]
Therefore, we get,
\[A-(B\cap C)\]\[=\]\[(A-B)\cup (A-C)\]
So the correct answer will be (4)
So, the correct answer is “Option 4”.

Note: Two sets A and B are said to be disjoint sets if the intersection of sets A and B gives us the null set. If we have to find the complement of the set then that set will be subtracted from the whole set that is the union of the set. If A is the subset of B, then all the elements of A will be in set B.