Combinational Logic Circuits

Minterm

A Minterm is a normal product term, in which every variable appears, and each variable only appears one time. There are 2ⁿ distinct minterms for a n-variable term. If we look a three-variable term, we have:

X	Y	Z	Product Term	Symbol
0	0	0	X'Y'Z'	m₀
0	0	1	X'Y'Z	m₁
0	1	0	X'YZ'	m₂
0	1	1	X'YZ	m₃
1	0	0	XY'Z'	m₄
1	0	1	XY'Z	m₅
1	1	0	XYZ'	m₆
1	1	1	XYZ	m₇

We can describe an Sum-Of-Products (actually, it is the Sum-Of-Minterms) expressions using minterms. (Since this is a SOPs, we do not care about any term that has a zero, because it does not contribute to the answer. If we have the expression:

F = X'Y'Z' + XYZ = m₀ + m₇

or we can use the Greek symbol Sigma for the logical sum (Boo lean OR) of the minterms.

The Sum-Of-Terms is obtained by the truth table. However, since minterms by definition must contain all variables, they need to be simplified and we call that result the Sum-Of-Products.

Maxterms

A maxterm is the complement of the minterm. We can describe a Product-Of-Sums expression using maxterms.

X	Y	Z	Sum Term	Symbol
0	0	0	X+Y+Z	M₀
0	0	1	X+Y+Z'	M₁
0	1	0	X+Y'+Z	M₂
0	1	1	X+Y'+Z'	M₃
1	0	0	X'+Y+Z	M₄
1	0	1	X'+Y+Z'	M₅
1	1	0	X'+Y'+Z	M₆
1	1	1	X'+Y'+Z'	M₇

If we have the expression:

F = (X + Y + Z)(X + Y + Z) = M₀ + M₇

or we can use the Greek symbol PI for the logical product (Boo lean AND) of the maxterms.

Complements

By inspection, you will see that:

M_j = m_j'

We can take it one step further.

Notice, that is m₁ + m₃ + m₄ + m₄ If we complement the expression, we get M₁ · M₃ · M₄ · M₄ This gives us:

This also works the other direction

m_j = M_j'

Maxterms are seldom used directly when dealing with Boolean functions, as we can replace them with the minterm list of F' and its truth table.

Combinational Logic Circuits

Minterm

Maxterms

Complements

© 2007 Gary L. Burt