Boolean Algebra: Canonical Forms

In Boolean algebra, any Boolean function can be expressed in a canonical form using minterms and maxterms. Canonical forms simplify using the laws of Boolean Algebra to minimize or other optimize digital circuit designs.

In the box below is additional terminology that is beyond the need of your course - but might aid further reseach into the maths involved.

Minterms are the logical AND of a set of variables (called 'product' . ), and maxterms are the logical OR of a set of variables (also called 'sum' +.

These terms are called duals because of their complementary-symmetry relationship as expressed by De Morgan's laws.

The dual canonical form of any Boolean function is a "sum of minterms" and a "product of maxterms."

The term "Sum of Products" or "SoP" is widely used for the canonical form that is a disjunction (OR) of minterms.

Its De Morgan dual is a "Product of Sums" or "PoS" for the canonical form that is a conjunction (AND) of maxterms.

So, how do we make use of canonical form?

a	b	c	output y
0	0	0	1
1	0	0	1
0	1	0	0
0	0	1	1
1	1	0	1
1	0	1	0
0	1	1	0
1	1	1	0

Suppose you are given a truth table for a circuit and asked to build it from logic gates.

The first thing you have to do is to write out an expression for the high output conditions.

Y will be high when any of the combinations of abc states result in a '1' in the truth table final column. The possible state combinations for a high output therefore have to be ORed together, while the state combinations themselves for inputs 'abc' have to be ANDed.

In a boolean equation the negative (or low) state is represented by an overscore (line over the top of a) symbol.

We now have to simplify this by using the Boolean Laws or Rules.

If we can get complementary pairs ORed together in a bracket we can eliminate them by replacing them with a 1 - which then disappears (identities) simplifying the expression a lot.

We therefore try to spot opportunities to do this.

c and c-bar can be extracted using the distribution laws, so can b and b-bar.