Canonical form (Boolean algebra)

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In a Boolean algebra, a Boolean function that is composed of standard logical operators can be expressed in a canonical form using the dual concepts of minterms and maxterms. All logical functions are expressible in canonical form, both as a "sum of minterms" and as a "product of maxterms". This allows for greater analysis into the simplification of these functions, which is of great importance in the minimization of digital circuits.

A Boolean function expressed as a disjunction (OR) of minterms is commonly known as the "sum of products", and its De Morgan dual is the "product of sums", which is a function expressed as a conjunction (AND) of maxterms.

Contents

1 Minterms

1.1 Indexing minterms
1.2 Functional equivalence

2 Maxterms

2.1 Dualization
2.2 Indexing maxterms
2.3 Functional equivalence

3 See also

Minterms

A minterm is a logical expression of n variables consisting of only the logical conjunction operator and the compliment operator.

For example, the following are examples of minterms:

a b'c

a' b c

There are 2ⁿ minterms of n variables - this is true since a variable in the minterm expression can either be in the form of itself or its complement - two choices per n variables.

Indexing minterms

In general, one assigns each minterm (ensuring the variables are written in the same order, usually alphabetic), an index based on the binary value of the minterm. A complemented term, like a' is considered a binary 0 and a noncomplemented term like a is considered a binary 1. For example, one would associate the number 6 with a b c'(110₂), and write the minterm expression as m₆. So m₀ of three variables is a'b'c'(000₂) and m₇ would be a b c(111₂).

Functional equivalence

It is apparent that minterm n gives a true value for the n+1 th unique function input for that logical function. For example, minterm 5, a b' c, is true only when a and c both are true and b is false - the input where a = 1, b = 0, c = 1 results in 1.

If one is given a truth table of a logical function, it is possible to write the function as a "sum of products". This is a special form of disjunctive normal form, qv. For example, if given the truth table

a b  f(a, b)
0 0  1
0 1  0
1 0  1
1 1  0

observing that the rows that have an output of 1 are the first and third, so we can write f as a sum of minterms m₀ and m₂.

If we wish to verify this:

f(a,b) = m₀ + m₂ = (a'b')+(ab')

then the truth table for this function, by direct computation, will be the same.

Maxterms

A maxterm is a logical expression of n variables consisting of only the logical disjunction operator and the complement operator. Maxterms are a dual of the minterm idea. Instead of using ANDs and complements, we use ORs and complements, and proceed similarly.

For example, the following are maxterms:

a+b'+c

a'+b+c

There are again 2ⁿ maxterms of n variables - this is true since a variable in the maxterm expression can also be in the form of itself or its complement - two choices per n variables.

Dualization

The complement of a minterm is the respective maxterm. This can be easily verified by using de Morgan's law. For example

m₁' = M₁

(a'b)' = a+b'

Indexing maxterms

Indexing maxterms is done in the opposite way as with minterms. One assigns each maxterm an index based on the order of its complements (again, ensuring the variables are written in the same order, usually alphabetic). For example, one might assign M₆ (Maxterm 6) to the maxterm a'+b'+c. Similarly M₀ of these three variables could be a+b+c and M₇ could be a'+b'+c'.

Functional equivalence

It is apparent that maxterm n now gives a false value for the n+1 th unique function input for that logical function. For example, maxterm 5, a'+b+c', is false only when a and c both are true and b is false - the input where a = 1, b = 0, c = 1 results in 0.

If one is given a truth table of a logical function, it is possible to write the function as a "product of sums". This a special form of conjunctive normal form, q.v. For example, if given the truth table

a b  f(a, b)
0 0  1
0 1  0
1 0  1
1 1  0

observing that the rows that have an output of 0 are the second and fourth, so we can write f as a product of maxterms M₁ and M₃.

If we wish to verify this:

f(a,b) = M₁ M₃ = (a+b')(a'+b')

then the truth table for this function, by direct computation, will be the same.

Canonical form (Boolean algebra)

Minterms

Indexing minterms

Functional equivalence

Maxterms

Dualization

Indexing maxterms

Functional equivalence

See also