Question

Let A = $\left\{ 1,2,3 \right\}$ , B = $\left\{ 2,3,4,5 \right\}$ and C = $\left\{ 3,4,5,6 \right\}$ be subsets of U = $\left\{ 1,2,3,4,5,6 \right\}$
Then, verify that:
A. (A’)’ = A
B. $\left( A\cup B \right)'=\left( A'\cap B' \right)$
C. $\left( A\cap B \right)'=\left( A'\cup B' \right)$

Answer 1

Hint: We will solve this question by first understanding the meaning of complement of a set, and then what is union and intersection of sets and after that we will prove all the given options.

Complete step by step answer:
Universal set: The set containing all objects or elements and of which all other sets are subsets.
Complement of a set: Complement of a set A, denoted by A’, is the set of all elements that belongs to the universal set but does not belong to set A.
Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .
Now,
A’ = U – A = $\left\{ 4,5,6 \right\}$
Now again taking complement we get,
(A’)’ = U – A’ = $\left\{ 1,2,3 \right\}$= A
Hence, we have proved the result (a).
Now, let us consider the next result,
$A\cup B=\left\{ 1,2,3,4,5 \right\}$
$\left( A\cup B \right)'=\left\{ 6 \right\}...........(1)$
$\begin{align}
  & A'=\left\{ 4,5,6 \right\} \\
 & B'=\left\{ 1,6 \right\} \\
\end{align}$
Now, we get
$A'\cap B'=\left\{ 6 \right\}..........(2)$
Hence we can see that (1) = (2).
$\left( A\cup B \right)'=\left( A'\cap B' \right)$
Therefore, we have proved result (b) also. Now, let us consider the last result to be proved.
$\begin{align}
  & A\cap B=\left\{ 2,3 \right\} \\
 & \left( A\cap B \right)'=\left\{ 1,4,5,6 \right\}..............(3) \\
\end{align}$
Now, we get
$A'\cup B'=\left\{ 1,4,5,6 \right\}...........(4)$
Hence (3) = (4),
$\left( A\cap B \right)'=\left( A'\cup B' \right)$
Therefore, (c) has been proved.

Note: We have used the definition of the given terms in the question to find the value of LHS and RHS of each option and proved that these two are equal, these definitions are very important and should be remembered.