Question

If $A,B$ and $C$are three finite sets, then what is $[(A\cup B)\cap C]'$ equal to?
A. \[A'\cup B'\cap C'\]
B. \[A'\cap B'\cap C'\]
C. \[A'\cap B'\cup C'\]
D. \[A\cap B\cap C\]

Answer 1

Hint: To solve this question, we will use the De-Morgan’s law. We will first apply the formula of the compliment of intersection of two sets on $(A\cup B)$ and $C$ in $[(A\cup B)\cap C]'$. Then we will apply the formula of compliment of the union of two sets on $A$ and $B$ and we will then get the value of $[(A\cup B)\cap C]'$.



Formula Used:De-Morgan’s law of compliment of intersection of two sets: $(A\cap B)'=A'\cup B'$.
De-Morgan’s law of compliment of union of two sets: $(A\cup B)'=A'\cap B'$



Complete step by step solution:We are given three finite sets $A,B$ and $C$ and we have to determine the value of $[(A\cup B)\cap C]'$.
We will first apply the De-Morgan’s Law of intersection of two sets in between $(A\cup B)$ and $C$ in$[(A\cup B)\cap C]'$.
$[(A\cup B)\cap C]'=(A\cup B)'\cup C'$
Now we will apply the De-Morgan’s law of union of two sets in between $A$ and $B$.
$\begin{align}
  & [(A\cup B)\cap C]'=(A\cup B)'\cup C' \\
 & =A'\cap B'\cup C'
\end{align}$



Option ‘C’ is correct



Note: The compliment of the intersection of two sets can be defined as the union of the compliment of both the sets while the compliment of the union of two sets can be defined as the intersection of the compliment of both the sets.
The compliment of any set can be defined as the set or collection of all the elements from the universal set which will not be present in that set. For example, if set A is collection of even numbers up to $5$ that is $A=\{2,4\}$, then its compliment will contain all the elements which is not present in $A=\{2,4\}$.