Question

If P, Q and R are subsets of A , then find the value of \[R \times {({P^c} \cup {Q^c})^c}\] .
A.\[\left( {R \times P} \right) \cap \left( {R \times Q} \right)\]
B. \[\left( {R \times Q} \right) \cap \left( {R \times P} \right)\]
C. \[\left( {R \times P} \right) \cup \left( {R \times Q} \right)\]
D. None of these.

Answer 1

Hint: Apply De-Morgan’s law of sets in the expression \[{P^c} \cup {Q^c}\], then calculate to obtain the required result.

Formula Used:
De-Morgan’s law of sets is,
\[{(P \cup Q)^c} = {P^c} \cap {Q^c}\]

Complete step by step solution:
The given expression is,
\[R \times {({P^c} \cup {Q^c})^c}\]
Apply De-Morgan’s law in the given expression.
\[R \times {\left[ {{{(P \cap Q)}^c}} \right]^c}\]
Now, apply that \[{\left( {{A^c}} \right)^c} = A\] .
\[R \times \left[ {(P \cap Q)} \right]\]
=\[\left( {R \times P} \right) \cap \left( {R \times Q} \right)\]\[\]

Hence the correct option is A.

Additional Information: A set that contains the elements present in the universal set but excluded from set A is referred to as the complement of set A. The definition of a union is the set created by combining components from two sets. The set of elements in A, B, or both is the result of adding sets A and B together. With the help of De Morgan’s law, we learn how mathematical expressions and concepts are related to their opposites. In set theory, De Morgan’s law deals with the union intersection of a set and its complements. It is simply an observation of the relation between the sets and their complements. With the help of De Morgan’s law, we can solve a problem easily.

Note: Students sometimes want to solve this type of problem by Venn diagram, but as the nature of P and Q are not given as the sets are disjoint or not it’s complicated to draw a Venn diagram with this information, rather than that student should use the De- Morgan’s law and solve it easily.