Solved Problem - Venn Diagrams
Question
Prove de Morgan's theorems
and
with the use of Venn diagrams
Answer
Considering the first expression above, the Venn Diagrams for
and 
are
respectively. The Boolean expression
.
requires the logical
AND of these two variables which, in terms of a Venn diagram, is given by that part
which is common to both diagrams. This is drawn as
For the right hand side of the expression first the logical OR of
and 
is required. In a Venn diagram representation, a logical OR is performed by taking
any shaded part of both of the Venn diagrams.
The three diagrams below denote ,
and 
respectively.
Finally we require the complement of OR
(i.e NOR ),
For a Venn diagram, complementing is
represented by all those parts of the Universe not populated by the original diagram.
Therefore is represented as
which is identical to the Venn diagram for .
.
In the case of the second expression above the approach is identical. Here
|
| + | | = |  |
| + | | = | + |
and
|
| . | | = |  |
| . | | = | . |
| and so |  | = | |
