Question

Q3. Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. Find A′, B′ , A′ ∩ B′, A ∪ B and hence show
that ( A ∪ B )′ = A′ ∩ B′.

Answer 1

Answer:

two methods

1st method

Explanation :

U = {1,2,3,4,5,6} , A={2,3} and B={3,4,5}

therefore ,

A'= {1,4,5,6}

B'={1,2,6}

A′ ∩ B′ = {1,6}

Now to show ( A ∪ B )′ = A′∩ B′ , first find ( A ∪ B )

(A U B) = {2,3,4,5,}

Now, ( A ∪ B )′ = {1,6}

As we have already know, A′ ∩ B′ = {1,6}

therefore, ( A ∪ B )′ = A′∩ B′

Additional information :

1] what is complement of set? :

consider A is the set, which is subset of the universal set U. then

the set of all those elements of U which do not belongs to A is called the complement of A with respect to U, it is denoted as A'

2]what is (AUB) [ union of sets ] ? :

if A and B are two sets, then the set of those elements which belong to A or to B or to both A and B, is called the union of the sets A and B,

it is denoted as AUB

3 ] what is intersection of sets? :

if A and B are two sets, then the set of those elements which belong to both A and B, i.e which are common to both A and B, is called the intersection of the set A and B

it is denoted by A∩B.

2nd method

Given Problem:

Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.

Find A′, B′ , A′ ∩ B′, A ∪ B and hence show that ( A ∪ B )′ = A′∩ B′.

Solution:

U = {1,2,3,4,5,6} , A={2,3} and B={3,4,5}

So,

A'= {1,4,5,6}

B' ={1,2,6}

A′ ∩ B′ = {1,6}

Now we have to represent that:

( A ∪ B )′ = A′∩ B′.

First of all we have to find that:

f( A ∪ B )

(A U B) = {2,3,4,5,6}

So,

(A ∪ B)′ = {1,6}

We know, A′ ∩ B′ = {1,6}

So,

Hence,

(A ∪ B)′ = A′∩ B′

Extra Information:

★ Set:

A set is a collectionof objects that are clearly identified. The objects in the set are called member & elements of the set. Each member is said to belong to the set. The symbol ∈ is used to denote belongs to or is an element of or is a member of.

★ Complement of a set:

If a set A is a subset of a given Universal Set (U), then the differnce set U/A is the complement  of A.

We write it as: U/A = A' , We say: A complement.

★ Universal set:

When a given set is overall set to which all the objects in a discussion belong, the given set is called the Universal Set.

We write is as: U, We say : Universal set.

★ Union of Sets:

The set which consits of all the elements that belong to A together with all the members of B is {0, 1, 3, 5, 7, 9} It is called the Union of A and B.

We can write is as :  A ∪ B