Minterms, Maxterms, Canonical and Standard Forms

Minterms:

Standard "products"

Contain each literal in the function in primed or unprimed form

For a function of N variables, there are 2^N minterms (one for each line in truth table)

For each line in a truth table of a function where that function is 1, the function is said to "contain" that minterm.

F(x,y,z) = xy + x'z

xyz

minterm

maxterm

F

[`F]
000

m₀ Û x'y'z'

M₀ Û (x+y+z)

0

1
001

m₁ Û x'y'z

M₁ Û (x+y+z')

1

0
010

m₂ Û x'yz'

M₂ Û (x+y'+z)

0

1
011

m₃ Û x'yz

M₃ Û (x+y'+z')

1

0
100

m₄ Û xy'z'

M₄ Û (x'+y+z)

0

1
101

m₅ Û xy'z

M₅ Û (x'+y+z')

0

1
110

m₆ Û xyz'

M₆ Û (x'+y'+z)

1

0
111

m₇ Û xyz

M₇ Û (x'+y'+z')

1

0

Maxterms:

Standard "sums"
Contain each literal in the function in primed or unprimed form
For a function of N variables, there are 2^N maxterms
A maxterm is the complement of the corresponding minterm
For each line in a truth table of a function where that function is 0, the function is said to "contain" that maxterm.

Canonical Forms

A listing of the minterms or maxterms of a function
Minterms - "Sum of Products" (SOP) form
Maxterms - "Product of Sums" (POS) form

For the function above,

F(CSOP) = m₁ + m₃ + m₆ + m₇ = å(1,3,6,7)

F(CSOP) = x'y'z + x'yz + xyz' + xyz

F(CPOS) = M₀ M₂ M₄ M₅ = Õ(0,2,4,5)

F(CPOS) = (x+y+z)(x+y'+z)(x'+y+z)(x'+y+z')

Standard Forms

A reduced version of the canonical forms
Minimal "Sum of Products" (MSOP) form - equation looks like a sum of products, and can not be algebraically reduced while still retaining that form
Minimal "Product of Sums" (MPOS) form - same as above, but looks like a sum of products

Examples:

MSOP:

xy+x'z

ABC + D

xy'z + x'z'
MPOS:

(x+y)(y+z')

(A+B'+C)(A'+C')

x(y+z)
Neither:

x(yz + w)

xy + z'(x'+w)

(x+y+z)(x+y'+z)

Algebraic Conversion to Canonical Forms

Conversion between standard forms

Complement of a function

File translated from T_EX by T_TH, version 0.9.