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If $A,B{\text{ and }}C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ then
(A) $A = C$
(B) $B = C$
(C) $A \cap B = \phi $
(D) $A = B$

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Last updated date: 20th Jun 2024
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Answer
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Hint: The union of two sets ‘X’ and ‘Y’ can be represented as $X \cup Y = X + Y - X \cap Y$ . Use this relation for two sets ‘A’ and ‘B’. Now utilize the given union equation $A \cup B = A \cup C$. Cancel out the same terms from both sides. Now use the intersection equation $A \cap B = A \cap C$in this equation to eliminate the intersection terms.

Complete step-by-step answer:
Here in this problem, we are given three sets $A,B{\text{ and }}C$ that follow the relation $A \cap B = A \cap C$ and $A \cup B = A \cup C$ . Using this information, we need to find which of the given four options is correct. We have four different equations in four options.
Before starting with the solution of the problem, we should understand a few concepts of set theory used here. The set operation intersection takes only the elements that are in both sets. The intersection contains the elements that the two sets have in common. The intersection is where the two sets overlap. The intersection of sets is denoted by $ \cap $ .
In set theory, the union of a collection of sets is the set of all elements in the collection. It is denoted by $ \cup $ . It is one of the fundamental operations through which sets can be combined and related to each other.
The union can be considered as the addition of the two sets with the subtraction of the intersection of these sets, which was added twice while adding. The union of two sets ‘X’ and ‘Y’ can be represented as:
$ \Rightarrow X \cup Y = X + Y - X \cap Y$ ………(i)
Now let’s consider the union equation, i.e. $A \cup B = A \cup C$
Using relation (i) in the above equation, we get:
$ \Rightarrow A \cup B = A \cup C \Rightarrow A + B + A \cap B = A + C + A \cap C$
Since set A is added on both sides of this equation, we can remove it from both sides. Thus we get:
$ \Rightarrow B + A \cap B = C + A \cap C$
Now we already know from the intersection equation that $A \cap B = A \cap C$.
Therefore, we can write it as:
$ \Rightarrow B + A \cap B = C + A \cap C \Rightarrow B + A \cap B = C + A \cap B \Rightarrow B = C$
So, we get a relation between set B and set C as: $B = C$
Hence, the option (B) is the correct answer.

Note: In questions of set theory, the use of concepts of union and intersection can play a useful role. An alternative approach for the same problem can be taken by assuming an element in one of the sets. Then we can establish two relations using the given equations. Also, the use of the Venn diagram can be a useful method to work on problems like this.