Minterms and Maxterms

// Minterms

For each combination of inputs to a logical fnction, there is an associated minterm.

A	B	C	Minterm
0	0	0	A'B'C'
0	0	1	A'B'C
0	1	0	A'BC'
0	1	1	A'BC
1	0	0	AB'C'
1	0	1	AB'C
1	1	0	ABC'
1	1	1	ABC

The minterm associated with each input combination is the AND, or product, of the input variables. Minterms and maxterms are useful for deriving Boolean equations from truth tables.

//Maxterms

For each combination of inputs to a logical function, there is an associated maxterm.

A	B	C	Maxterm
0	0	0	A+B+C
0	0	1	A+B+C'
0	1	0	A+B'+C
0	1	1	A+B'+C'
1	0	0	A'+B+C
1	0	1	A'+B+C'
1	1	0	A'+B'+C
1	1	1	A'+B'+C'

The maxterm associated with each input combination is the OR, or sum, of the input variables. Maxterms and minterms are useful for deriving Boolean equations from truth tables.

// SOP- Sum-of-Products

There are two commonly used techniques for deriving Boolean equations from a truth table. In the first technique, the minterms corresponding to each line in the truth table for which the output is logic 1 are extracted and combined using | OR operators. This method results in an equation said to be in the sum-of-products form. In the second technique, the maxterms corresponding to each line in the truth table for which the loutput is logic 0 are combined using & AND operators. This method results in an equation said to be in product-of-sums form.

A	B	C	F
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	1
1	0	1	1
1	1	0	0
1	1	1	0

SOP expression=A'B'C+A'BC+AB'C'+AB'C

For functions whose output is logic 1 fewer times than it is logic 0, it is generally easier to extract a sum-of-products equation. The sum-of-products and products of sums equations complement each other and return identical results. An equation in either form can be transformed into its alternative form by means of the appropriate DeMorgan Transformation

// POS- Product-of-Sums

Find the logic 0 outputs and invert those inputs in the product of sum format:

A	B	C	F
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	1
1	0	1	1
1	1	0	0
1	1	1	0

POS expression=(A+B+C)(A+B'+C)(A'+B'+C)(A'+B'+C')

// DeMorgan Transformations

Augustus DeMorgan(1806-1871) described a set of rules which facilitate the conversion of Boolean expressions into alternative and often more convenient forms.