Canonical Forms

The functionality of any logic circuit can be expressed in one of two alternative and equivalent canonical forms. These canonical forms consist of a Boolean algebraic expression. They are generally developed from a truth table. They are

The minterm form
Known as the first canonical form, this a pure OR combination of minterms where a minterm is an AND function that includes each variable once in its normal or complemented form. The first canonical form is also known as the sum of products.
The maxterm form
Known as the second canonical form, this a pure AND combination of maxterms where a maxterm is an OR function that includes each variable once in its normal or complemented form. The second canonical form is also known as the product of sums.

The construction of these forms is best illustrated by the examples below.

The Minterm Form

Consider the truth table below for the output Y from a combinatorial logic circuit comprising three inputs, A, B and C

A	B	C	Y
1	1	1	1
1	1	0	0
1	0	1	0
1	0	0	1
0	1	1	0
0	1	0	0
0	0	1	0
0	0	0	0

The first canonical form is developed from the output 1's in the truth table. As can be seen, Y is only 1 for the 1st and 4th rows of the truth table. Therefore the minterm (AND function) expressions for these two rows are formed and OR-ed together to give the minterm form for the circuit as

The corresponding circuit would be implemented with AND-OR logic i.e. with the outputs from one or more AND gates being OR-ed together to give the final output.

The Maxterm Form

Consider the truth table below for the output Y from a combinatorial logic circuit comprising three inputs, A, B and C

A	B	C	Y
1	1	1	0
1	1	0	1
1	0	1	1
1	0	0	0
0	1	1	1
0	1	0	1
0	0	1	1
0	0	0	1

The second canonical form is developed from the output 0's in the truth table. As can be seen, Y is only 0 for the 1st and 4th rows of the truth table. Developing the maxterm expression here is slightly more complicated and there are two approaches.

In the first approach we first develop the minterm expression for the output 0's (not 1's) in the truth table. For the truth table above this will be given by

Then it is necessary to apply the rules of Boolean algebra for converting minterm expressions to maxterm expressions as is described in the Boolean Algebra summary. This leads to the final maxterm form for this truth table of

The second approach allows the maxterm form to be derived directly from the output 0's in the truth table using the following rules.

Take each line in the truth table where the output is 0 and
Invert the variables (e.g. if is 1 then write etc.)
OR these variables together to form the maxterm
build the second canonical form from the AND of these maxterms

In the case of the truth table above it is possible to go directly to the final maxterm form using this approach.

The corresponding circuit would be implemented with OR-AND logic i.e. with the outputs from one or more OR gates being AND-ed together to give the final output.

PHY107 Home