Question

Let $U=\left\{ 1,2,3,4,5,6 \right\}$, $A=\left\{ 2,3 \right\}$ and $B=\left\{ 3,4,5 \right\}$. Show that $\left( A\cup B \right)'=A'\cap B'$.

Answer 1

Hint: From the question given we have to prove that $\left( A\cup B \right)'=A'\cap B'$ from the given sets. To prove this first we will find the set of $\left( A\cup B \right)'$, and the set of $A'\cap B'$. Then we will compare these two sets if the elements in these two sets are equal then we can prove this.

Complete step by step solution:
From the question given we have to show that,
$\Rightarrow \left( A\cup B \right)'=A'\cap B'$
In the question we have been given the three sets they are,
$\Rightarrow U=\left\{ 1,2,3,4,5,6 \right\}$
$\Rightarrow A=\left\{ 2,3 \right\}$
$\Rightarrow B=\left\{ 3,4,5 \right\}$
First, we will find the set of A union B that is $A\cup B$. We will get,
$\Rightarrow A\cup B=\left\{ 2,3 \right\}\cup \left\{ 3,4,5 \right\}$
$\Rightarrow A\cup B=\left\{ 2,3,4,5 \right\}$
From this we will get the set of $\left( A\cup B \right)'$, we will get,
$\Rightarrow \left( A\cup B \right)'=U-\left( A\cup B \right)$
$\Rightarrow \left( A\cup B \right)'=\left\{ 1,2,3,4,5,6 \right\}-\left\{ 2,3,4,5 \right\}$
$\Rightarrow \left( A\cup B \right)'=\left\{ 1,6 \right\}$
Therefore, the $\left( A\cup B \right)'$ set is $\left\{ 1,6 \right\}$
Now to find $A'\cap B'$, first we have to find the individually $A'$ and $B'$.
First, we will find $A'$ is
$\Rightarrow A'=U-A$
$\Rightarrow A'=\left\{ 1,2,3,4,5,6 \right\}-\left\{ 2,3 \right\}$
$\Rightarrow A'=\left\{ 1,4,5,6 \right\}$
Now, we will find $B'$ is,
$\Rightarrow B'=U-B$
$\Rightarrow B'=\left\{ 1,2,3,4,5,6 \right\}-\left\{ 3,4,5 \right\}$
$\Rightarrow B'=\left\{ 1,2,6 \right\}$
Therefore, now the intersection of $A'$ and $B'$ that is $A'\cap B'$ is,
$\Rightarrow A'\cap B'=\left\{ 1,6 \right\}\cap \left\{ 1,2,6 \right\}$
$\Rightarrow A'\cap B'=\left\{ 1,6 \right\}$
Therefore, by comparing the $\left( A\cup B \right)'$ and $A'\cap B'$ they both are equal that is,
$\Rightarrow \left( A\cup B \right)'=A'\cap B'=\left\{ 1,6 \right\}$
Hence proved.

Note: Students should know the concept of sets and students should recall all the symbols, formulas, concept of sets before doing this problem. Students should know the meaning of symbols if they are confused in symbols the whole answer will be wrong. The symbol $\cap $ represents the intersection, and the symbol $\cup $represents the union means combine.