Question

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

Answer 1

To verify De Morgan's law of complementation:

$(A\cup B)\text{'}=A\text{'}\cap B\text{'}$

The highlighted or the green coloured portion denotes$A\cup B$. The complement of the union of $A$and $B$, i.e., $(A\cup B)’$ is a set of all those elements which are not in $A\cup B$. 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, $B\text{'}$ is represented as:

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

If fig.$3$and $4$ 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 $B$.

$(A\cup B)\text{'}=A\text{'}\cap B\text{'}$

Hence, proved.