Conjunctive Normal Form

Definition

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

It has this shape:

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

In short:

AND of OR-groups

Conjunctive normal form organizes logic as clauses joined by AND.

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Clause

In conjunctive normal form, each OR-group is called a clause.

Example:

(P OR Q)

Another clause:

(NOT R OR S)

Full expression:

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

This is in conjunctive normal form.

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

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Why This Form Matters

Normal forms make logical expressions easier to compare and process.

Instead of many possible shapes, we use one standard shape.

Example:

(P OR Q) AND (R OR S)

This structure makes it clear:

both clauses must hold

and within each clause:

at least one part must hold

Normal form reduces structural variation.

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Satisfaction

A conjunctive normal form expression is true only when every clause is true.

Example:

(P OR Q) AND (R OR S)

For the full expression to be true:

P OR Q must be true
R OR S must be true

Each clause allows alternatives.

But all clauses must be satisfied.

CNF requires every clause to be satisfied.

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Connection To AND And OR

Conjunctive means joined by AND.

Normal form means a standardized shape.

So:

Conjunctive normal form

means:

standard form built as AND of clauses

where each clause is built using OR.

CNF layers OR inside AND.

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Final Summary

• Conjunctive normal form is an AND of OR-groups.
• Each OR-group is a clause.
• The full expression is true only when every clause is true.
• CNF gives logic a standard structure.
• CNF is useful because it makes symbolic expressions easier to compare, transform, and check.