Full conjunction

A full conjunction (also Minterm or elementary conjunction ) is a special conjunction term in propositional logic , i.e. H. a number of literals ( Boolean variables ), all of which are linked by a logical and ( ). All variables of the considered -digit Boolean function must appear in the conjunction term. Full conjunctions can be combined to form a disjunctive normal form, for example using the Quine and McCluskey method . ${\displaystyle \wedge }$ ${\displaystyle n}$ ${\displaystyle n}$

Examples

Examples of 3-digit Boolean functions

• ${\displaystyle e_{1}\wedge e_{2}\wedge e_{3}}$
• ${\displaystyle e_{1}\wedge \neg e_{2}\wedge e_{3}}$
• ${\displaystyle \neg e_{1}\wedge e_{2}\wedge \neg e_{3}}$

Standard numbering of the full conjunctions

Full conjunctions can be numbered naturally. Think of the variables in a row, e.g. B. . If the respective literal occurs negated for a concrete full conjunction , it is replaced by a 0, otherwise by a 1. A binary number is created that can be interpreted in decimal form. This decimal number is known as the number or index of the minterm. If you want to designate this minterm via its index , you write . The same goes for the maxterms for disjunctions. ${\displaystyle X_{n}X_{n-1}...X_{2}X_{1}}$ ${\displaystyle X_{i}}$ ${\displaystyle i}$ ${\displaystyle m_{i}}$ ${\displaystyle M_{i}}$

Comparison of Minterm / Maxterm

The following table shows the difference between the Maxterm and Minterm representation:

index	${\displaystyle x_{2}}$	${\displaystyle x_{1}}$	${\displaystyle x_{0}}$	Minterm	Max term
0	0	0	0	${\displaystyle \neg x_{2}\wedge \neg x_{1}\wedge \neg x_{0}}$	${\displaystyle x_{2}\vee x_{1}\vee x_{0}}$
1	0	0	1	${\displaystyle \neg x_{2}\wedge \neg x_{1}\wedge x_{0}}$	${\displaystyle x_{2}\vee x_{1}\vee \neg x_{0}}$
2	0	1	0	${\displaystyle \neg x_{2}\wedge x_{1}\wedge \neg x_{0}}$	${\displaystyle x_{2}\vee \neg x_{1}\vee x_{0}}$
3	0	1	1	${\displaystyle \neg x_{2}\wedge x_{1}\wedge x_{0}}$	${\displaystyle x_{2}\vee \neg x_{1}\vee \neg x_{0}}$
4th	1	0	0	${\displaystyle x_{2}\wedge \neg x_{1}\wedge \neg x_{0}}$	${\displaystyle \neg x_{2}\vee x_{1}\vee x_{0}}$
5	1	0	1	${\displaystyle x_{2}\wedge \neg x_{1}\wedge x_{0}}$	${\displaystyle \neg x_{2}\vee x_{1}\vee \neg x_{0}}$
6th	1	1	0	${\displaystyle x_{2}\wedge x_{1}\wedge \neg x_{0}}$	${\displaystyle \neg x_{2}\vee \neg x_{1}\vee x_{0}}$
7th	1	1	1	${\displaystyle x_{2}\wedge x_{1}\wedge x_{0}}$	${\displaystyle \neg x_{2}\vee \neg x_{1}\vee \neg x_{0}}$

Realization of decoder circuits with minter terms / max terms:

	Minterm	Max term
0	NOR gate	AND gate
1	OR gate	NAND gate

Designations

Minterme

• A single minterm:
  • For exactly one assignment of function value 1
  • Minimality:
    • maximum number of zeros
    • minimum number of ones

(apart from the trivial null function)

Maxterms

• A single maxterm:
  • Function value 0 for exactly one assignment
  • Maximality:
    • maximum number of ones
    • minimum number of zeros

(apart from the trivial one function)

Relation to the Karnaugh-Veitch diagram

One also speaks of the minterm of a function ${\displaystyle F}$ if this implies , i. H. if applies ${\displaystyle F}$

${\displaystyle M(X)=1\Rightarrow F(X)=1}$.

Here is the vector of the input variable. Such minterms , in a reversible manner, unambiguously correspond to those fields of a Karnaugh-Veitch diagram which contain the value 1 for the function under consideration. ${\displaystyle X}$ ${\displaystyle M}$