Question

If the universal set $U=\left\{ 1,2,3,4,5,6,7 \right\}$ , $A=\left\{ 1,2,5,7 \right\}$ , $B=\left\{ 3,4,5,6 \right\}$. Verify $\left( A\cup B \right)'=A'\cap B'$.

Answer 1

Hint: Using sets A, B then find $A\cup B$ after eliminating all elements of $A\cup B$ from the universal set to find $\left( A\cup B \right)'$. Now find $A',B'$ and then find $A'\cap B'$ and then equate it with $\left( A\cup B \right)'$ .

Complete step-by-step answer:

We are provided with three sets which are $U=\left\{ 1,2,3,4,5,6,7 \right\}$, $A=\left\{

1,2,5,7 \right\}$ and $B=\left\{ 3,4,5,6 \right\}$ where U is a universal set.

At first, we briefly understand what is set.

In mathematics sets is a well defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. For example, the number 2, 4, 6 are distinct and considered separately, but they are considered collectively they form a single set of single set of size three written as $\left\{ 2,4,6 \right\}$ which could also be written as $\left\{ 2,4,6 \right\}$.

Here Universal set U means the number of elements contained in U is the maximum number of elements any other set can have.

Here $A=\left\{ 1,2,5,7 \right\}$ and $B=\left\{ 3,4,5,6 \right\}$

So, at first we will find $A\cup B$ which means the set of $A\cup B$ should contain all the elements of both A and B.

So, $\left( A\cup B \right)=\left\{ 1,2,3,4,5,6,7 \right\}$

Then we are asked to find $\left( A\cup B \right)'$ which means that we have to take all those elements after eliminating all the elements of the $A\cup B$ from universal set U.

As we know $U=\left\{ 1,2,3,4,5,6,7 \right\}$ and $A\cup B=\left\{ 1,2,3,4,5,6,7 \right\}$

So, $\left( A\cup B \right)'=\phi $ as it contains no elements.

Now, we will find the $A'\cap B'$ element.

At first we will find A’ so, $A=\left\{ 1,2,5,7 \right\}$ and $U=\left\{ 1,2,3,4,5,6,7 \right\}$
So, $A'=\left\{ 3,4,6 \right\}$ which we got by removing all the elements of A from U.

Now, we will go for B’ so, $B=\left\{ 3,4,5,6 \right\}$ and $U=\left\{ 1,2,3,4,5,6,7 \right\}$
So, $B'=\left\{ 1,2,7 \right\}$ which we got by eliminating elements of B from U.

Now, as we know $A'=\left\{ 3,4,6 \right\}$ and $B'=\left\{ 1,2,7 \right\}$ and we have to find $A'\cap B'$ which is $\phi $ because as there is no common elements in common with A’ and B’ .

So, $\left( A\cup B \right)'=\phi =\left( A'\cap B' \right)$

Hence Proved.

Note: One can also do by drawing Venn diagrams and finding elements of each set such as $A',B',A\cup B,A'\cap B',\left( A\cup B \right)'$ and substitute to prove it.