Question

Verify

{(A \cup B)}^{'} = A^{'} \cap B^{'}

using Venn diagrams.

Accepted Answer

Given to verify: ${(A \cup B)}^{'} = A^{'} \cap B^{'}$ using Venn diagram

We know that,

Venn diagram are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way

Let $U$ be the universal set containing finite sets $A a n d B$ , then ${(A \cup B)}^{'} = A^{'} \cap B^{'}$ is called De Morgan's law for complementation.

The given law can verify using Venn diagram as shown below:

......... $(I)$

....... $(I I)$

From $(I) a n d (I I)$ ${(A \cup B)}^{'} = A^{'} \cap B^{'}$ is verified

Let us consider an example:

$U = \{0, 1, 2, 3, 4, 5, 6\}, A = \{1, 3, 5\}, B = \{0, 3, 4, 5\}$

The union of sets $A a n d B$ is the set of all elements which are either in $A$ or $B$ or both.

$A \cup B = \{1, 3, 5\} \cup \{0, 3, 4, 5\}$

$A \cup B = \{0, 1, 3, 4, 5\}$

The complement of a set $A$ is the set of all elements of $U$ that are not in $A$

${(A \cup B)}^{'} = U - (A \cup B)$

$= \{2, 6\}$ ..... $(1)$

The intersection of two sets $A a n d B$ is the set of all elements which are common to both $A a n d B$

$A^{'} = U - A$

$= \{0, 2, 4, 6\}$

$B^{'} = U - B$

$= \{1, 2, 6\}$

$A^{'} \cap B^{'} = \{0, 2, 4, 6\} \cap \{1, 2, 6\}$

$= \{2, 6\}$ ....... $(2)$

From $(1)$ and $(2)$ , it is verified that ${(A \cup B)}^{'} = A^{'} \cap B^{'}$ .

Tamil Nadu Board Solutions for Chapter: Set Language, Exercise 25: Solved Example

Tamil Nadu Board Mathematics Solutions for Exercise - Tamil Nadu Board Solutions for Chapter: Set Language, Exercise 25: Solved Example

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