Disjunctive Normal Form

Definition

Disjunctive normal form is a standard way to write logical expressions.

It has this shape:

(A AND B) OR (C AND D) OR (E AND F)

In short:

OR of AND-groups

Disjunctive normal form organizes logic as alternatives joined by OR.

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Term

In disjunctive normal form, each AND-group is often called a term.

Example:

(P AND Q)

Another term:

(NOT R AND S)

Full expression:

(P AND Q) OR (NOT R AND S)

This is in disjunctive normal form.

A term is an AND-group inside the larger OR structure.

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Why This Form Matters

Disjunctive normal form describes different ways a statement can be true.

Example:

(P AND Q) OR (R AND S)

This says the expression is true if:

P and Q are both true

or:

R and S are both true

Each term is one satisfying path.

DNF describes alternative sufficient cases.

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Satisfaction

A disjunctive normal form expression is true if at least one term is true.

Example:

(P AND Q) OR (R AND S)

For the full expression to be true, either:

P AND Q

is true, or:

R AND S

is true.

DNF needs at least one complete term to be satisfied.

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CNF vs DNF

Conjunctive normal form:

AND of OR-groups

Disjunctive normal form:

OR of AND-groups

CNF emphasizes all constraints that must be satisfied.

DNF emphasizes alternative cases that make the expression true.

CNF is all clauses; DNF is any complete case.

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Final Summary

• Disjunctive normal form is an OR of AND-groups.
• Each AND-group can be called a term.
• DNF describes alternative ways an expression can be true.
• A DNF expression is true if at least one term is true.
• DNF is different from CNF.
• DNF gives logical expressions a standard alternative-case structure.