Use Venn Diagrams To Verify Demorgans Law Of Complementation.

Sol:

Given:\( (A \cup B)’ = A’ \cup B'\)

 Union of Sets

The highlighted or the green coloured portion denotes A∪B. The complement of the union of A and B, i.e., (A∪B)’is a set of all those elements which are not in A∪B. This can be visualised as follows:

Complement of sets

Similarly, R.H.S of equation one can be represented using Venn Diagrams as well; the first part, i.e., A’ can be depicted as follows:

Complement of set A

The portion in black indicates set A and the blue part denotes its complement, i.e., A’.

Similarly, B’ is represented as:

 

Complement of set B

 

The portion in black indicates set B and the yellow part denotes its complement, i.e., B’.

If fig. 3 and 4 are superimposed on one another, we get a figure similar to the complement of sets.

Intersection of complements of sets

According to De Morgan’s first law, the complement of the union of two sets A and B equals the intersection of the complement of the sets A and B.

\((A \cup B)’ = A’ \cap B'\)

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