Question

The set $\left( A\cup B\cup C \right)\cap \left( A\cap B'\cap C' \right)'\cap C'$ is equal to
\[\begin{align}
  & A.B\cap C' \\
 & B.A\cap C \\
 & C.B\cap C' \\
 & D.A\cap C' \\
\end{align}\]

Answer 1

Hint: In solving these type of questions, we will use various properties of set to reach the final simplified answer. Properties which we will use are:
(i) $\left( A\cup B \right)'=A'\cap B'$ which is called De Morgan's first law.
(ii) $\left( A\cap B \right)'=A'\cup B'$ which is called De Morgan's second law.
(iii) $\left( A\cup B \right)\cup C=A\cup \left( B\cup C \right)$ which is called association law.
(iv) $\left( A\cup B \right)\cap \left( A\cup C \right)=A\cup \left( B\cap C \right)$ which is called distributive law.
(v) $\left( A\cup B \right)=\left( B\cup A \right)$ which is called commutative law.
(vi) $A\cup \varnothing =A$ which is called law of identity element.
(vii) $A\cap A'=\varnothing $ which is called complement property.

Complete step-by-step answer:
We are given set $\left( A\cup B\cup C \right)\cap \left( A\cap B'\cap C' \right)'\cap C'$
Using distributive law, we will solve first two terms together and then third term.
\[\left[ \left( A\cup B\cup C \right)\cap \left( A\cap B'\cap C' \right)' \right]\cap C'\]
We will use distributive law in both terms to combine $B\cup C\text{ and }B'\cap C'$ we get:
\[\left[ \left( A\cup \left( B\cup C \right) \right)\cap \left( A\cap \left( B'\cap C' \right) \right)' \right]\cap C'\]
Let us use De Morgan law in second term, so we get:
\[\left[ \left( A\cup \left( B\cup C \right) \right)\cap \left( A'\cap \left( B\cap C \right) \right) \right]\cap C'\]
Using commutative rule, we get:
\[\left[ \left( \left( B\cup C \right)\cup A \right)\cap \left( \left( B\cup C \right)\cup A' \right) \right]\cap C'\]
Using distributive law, we get:
\[\left[ \left( B\cup C \right)\cup \left( A\cap A' \right) \right]\cap C'\]
As we know, $A\cap A'=\varnothing $ using complement property, therefore, equation becomes
\[\left[ \left( B\cup C \right)\cup \varnothing \right]\cap C'\]
As we know, $A\cup \varnothing =A$ using law of identity element, we get:
\[\left[ B\cup C \right]\cap C'\]
Using distributive law we get:
\[\left( B\cap C' \right)\cup \left( C\cap C' \right)\]
As we know $C\cap C'=\varnothing $ using complement property, therefore, equation becomes
\[\left( B\cap C' \right)\cup \varnothing \]
Since, $A\cup \varnothing =A$ using law of identity element we get:
\[\left( B\cap C' \right)\cup \varnothing =B\cap C'\]
Hence, $\left( A\cup B\cup C \right)\cap \left( A\cap B'\cap C' \right)'\cap C'$ is equal to $B\cap C'$

So, the correct answer is “Option C”.

Note: Students should remember all the properties of sets. Notation ‘$^{C}$’ can also be used for representing the complement of a set instead of ‘$'$’ only.
Students should not get confused with $\cup ,\cap $ as both are completely different signs.