Maxterm and Minterm, Canonical Forms (1.11)

• Minterm: For n variables, the minterm is a product (•, AND) term that contains each variable exactly once, in complemented or uncomplemented form.
• In minterm mj, a variable is complemented if its value in the binary equivalent of j is 0.
•  Maxterm: For n variables, the maxterm is a sum (+, OR) term which contains each variable exactly once, in complemented or uncomplemented form.
• In maxterm Mj, a variable is complemented if its value in the binary equivalent of j is 1.
• Truth Table notation for minterms, maxterms

 Minterms and Maxterms are easy to denote using a truth table.

 Example (3 variables):

Canonical Forms

• Any Boolean function f( ) can be expressed as a  unique sum of minterms (except for commutativity).
• The minterms included are those m_j such that f( ) = 1 in row j of the truth table for f( ).
• Any Boolean function f( ) can be expressed as a unique product of maxterms (except for  commutativity).
• The maxterms included are those Mj such that  f( ) = 0 in row j of the truth table for f().

Example:  Truth table for f1(a,b,c):

• The canonical sum-of-products form for f1 is f1(a,b,c) = a’b’c + a’bc’ + ab’c’ + abc’
• The canonical product-of-sums form for f1 is f1(a,b,c)= (a+b+c)•(a+b’+c’)•(a’+b+c’)•(a’+b’+c’).