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Question

Use venn diagrams To verify De Morgan's law of complementation.


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Solution

To verify De Morgan's law of complementation:

(AB)'=A'B'

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

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:

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

Similarly, B' is represented as:

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

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

(AB)'=A'B'

Hence, proved.


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