Question

For any two sets A and B, if $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for some set X, then
A). $A - B = A \cap B$
B). $A = B$
C). $B - A = A \cap B$
D). None of the above

Answer 1

Here we are asked to find the relation between two sets A and B. We are given some data using that and some properties and laws of sets we will find the relation between sets A and B. We will start by applying the absorption law to both sets A and B. Then we will simplify it by using the given data.
Formula: Absorption law: For any set two sets $A$ and $B$, $A \cap \left( {A \cup B} \right) = A \cup \left( {A \cap B} \right) = A$
Distributive law: For all sets $A,B$ and $C$, $A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ and $A \cup \left( {B \cap C} \right) = \left( {A \cup B} \right) \cap \left( {A \cup C} \right)$

Complete step-by-step solution:
It is given that $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for any two sets $A$and$B$.
Let us consider set $A$ and $X$ then by absorption law we know that for any set two sets $A$and$B$, $A \cap \left( {A \cup B} \right) = A \cup \left( {A \cap B} \right) = A$
Thus, here for the sets $A$ and $X$ we get$A = A \cap \left( {A \cup X} \right)$.
From the given data we have $A \cup X = B \cup X$ let us substitute it in the above expression.
A = A \cap \left( {A \cup X} \right)$$$$ \Rightarrow A = A \cap \left( {B \cup X} \right)
Now let us expand the above expression using the distributive law.
We know that from distributive law, for all sets $A,B$ and$C$, $A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$
Here the sets are $A,B$and$X$. Thus, we get
$\Rightarrow A = \left( {A \cap B} \right) \cup \phi$
Again, from the given data we have$A \cap X = \phi$. Substituting this in the above expression we get
$\Rightarrow A = \left( {A \cap B} \right) \cup \phi$
$\Rightarrow A = \left( {A \cap B} \right)$……………………..$(1)$
Now let us consider the sets $B$ and $X$ then by absorption law we know that for any set two sets $A$and$B$, $A \cap \left( {A \cup B} \right) = A \cup \left( {A \cap B} \right) = A$
Thus, for the sets $B$and $X$ we get$B = B \cap \left( {B \cup X} \right)$.
From the given data we have $A \cup X = B \cup X$ let us substitute it in the above expression.
B = B \cap \left( {B \cup X} \right)$$$$ \Rightarrow B = B \cap \left( {A \cup X} \right)
Now let us expand the above expression using the distributive law.
We know that from distributive law, for all sets $A,B$ and$C$, $A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$
Here the sets are $A,B$and$X$. Thus, we get
$\Rightarrow B = \left( {B \cap A} \right) \cup \left( {B \cap X} \right)$
Again, from the given data we have$B \cap X = \phi$. Substituting this in the above expression we get
$\Rightarrow B = \left( {B \cap A} \right) \cup \phi$
$\Rightarrow B = \left( {B \cap A} \right)$
$\Rightarrow B = \left( {A \cap B} \right)$…………………. $(2)$
Now we have two expressions $(1)$and $(2)$ comparing these two expressions we get
A = \left( {A \cap B} \right) = B$$$$ \Rightarrow A = B
Thus, we got that the two sets A and B are equal. Now let us see the options to find the correct answer.
Option (a) $A - B = A \cap B$ is an incorrect answer since if A and B are equal their difference will be zero.
Option (b) $A = B$ is the correct answer as we got the same relation from our calculation above.
Option (c) $B - A = A \cap B$ is an incorrect answer since if A and B are equal their difference will be zero.
Option (d) None of these is an incorrect option as we got option (b) as a right answer.
Hence, option (b) $A = B$ is the correct answer.

Note: Here at first, we have used the absorption law because applying that law results in giving a term that is already given in the question which will be easier for us to simplify it further. So, whenever we derive some result, it is necessary to use the appropriate law or property which makes us use the data we already have.