Question

Use Venn diagrams to verify the following two relationships for any events Aand B (these are called De Morgan’s laws):

a.\(\left( {A \cup B} \right)' = A' \cap B'\)

b.\(\left( {A \cap B} \right)' = A' \cup B'\)

Hint:In each part, draw a diagram corresponding to the left side and another corresponding to the right side.)

Short answer

a. Venn diagram for the given relationship is:

b. Venn diagram for the given relationship is:

Step-by-step solution

Given information

De Morgan’s laws are provided.

Verifying first relationship using Venn diagrams

a.

The law is provided as,

\(\left( {A \cup B} \right)' = A' \cap B'\)

For L.H.S,

The Venn diagram for the event \(\left( {A \cup B} \right)\) is represented as,

Fig(1)

Therefore, using the above diagram, the Venn diagram for\(\left( {A \cup B} \right)'\)is represented as,

Fig(2)

For R.H.S,

The Venn diagram for the event \(A'\) is represented as,

Fig(3)

The Venn diagram for\(B'\) is represented as,

Fig(4)

Therefore, the intersection of the above two diagrams gives\(A' \cap B'\)and is represented as,

Fig(5)

Therefore, it can be observed that Fig(2) and Fig (5) are the same.

Hence verified.

Verify the second relationship using Venn diagrams

b.

The law is provided as,

\(\left( {A \cap B} \right)' = A' \cup B'\)

For L.H.S,

The Venn diagram for the event \(\left( {A \cap B} \right)\) is represented as,

Fig (6)

Therefore, using the above diagram, the Venn diagram for \(\left( {A \cap B} \right)'\)is represented as,

Fig (7)

For R.H.S,

The Venn diagram for the event \(A'\) is represented as,

Fig(8)

The Venn diagram for \(B'\)is represented as,

Fig(9)

Therefore, the union of the above two diagrams gives\(A' \cup B'\)and is represented as,

Fig(10)

Therefore, it can be observed that Fig(7) and Fig (10) are the same.

Hence verified.