Question

Let A, B and C be the sets such that $A \cup B = A \cup C{\text{ and A}} \cap {\text{B = A}} \cap C$ . Show that B=C .

Answer 1

In this particular type of question we need to use the general formula of set i.e. $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$. With this we will be using some of the other basic formulas of set operations. Firstly we will consider $A \cup B = A \cup C$ then we will take the intersection of B and C individually and we will take the help of above formulas. Then we will get two equations. After comparing these two equations, we will get our answer.

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Complete step-by-step answer:**

We have,

$A \cup B = A \cup C{\text{ and A}} \cap {\text{B = A}} \cap C$

Let’s consider $A \cup B = A \cup C$

Take the intersection of C on both sides. We get,

$\Rightarrow (A \cup B) \cap C = (A \cup C) \cap C$

But we know that,

$(A \cup C) \cap C = C$ and $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$

We will put these two formulas in the above equation.

We get,

$(A \cap C) \cup (B \cap C) = C$

This can be written as,

$(A \cap B) \cup (B \cap C) = C$ ……………(1) $(\therefore A \cap C = A \cap B)$

Again $A \cup B = A \cup C$

Now take the intersection of B on both sides. We get

$\Rightarrow (A \cup B) \cap B = (A \cup C) \cap B$

We know that,

$(A \cup B) \cap B = B$ and $(A \cup C) \cap B = (A \cap B) \cup (C \cap B)$

We will put these two formulas in the above equation.

We get,

$B = (A \cap B) \cup (C \cap B)$

$\therefore (A \cap B) \cup (C \cap B)=B$

This can be written as,

$(A \cap B) \cup (B \cap C)=B$ ……………(2) $(\therefore B \cap C= C \cap B)$

From the equation (1) and (2), we can see that LHS are the same in both cases.

So we can conclude that our RHS will also be the same.

i.e. B = C

Hence proved.

Note: In these questions we need to recall the general addition formula of two sets which are in union or intersection . We can also use a Venn diagram to get to the required results . Note that in a Venn diagram with sets A and B intersecting with each other if we add sets A and B , we will get the common element of the two sets which we need to subtract once , this is what a general addition rule tells us.