DNF types (Disjunctive Normal Form) in PHP - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is Disjunctive Normal Form (DNF) in logic?

DNF is a way to write logical expressions as an OR of ANDs. Each AND group contains only variables or their negations, and the whole expression is a combination of these groups connected by OR.

beginner

How can DNF be useful in programming?

DNF helps simplify complex logical conditions by breaking them into simpler parts. This makes it easier to evaluate, optimize, or convert logic into code like PHP if-statements.

beginner

In PHP, how would you represent a DNF expression like (A AND B) OR (NOT C AND D)?

You can use boolean operators: ($A && $B) || (!$C && $D). This matches the DNF structure of OR between AND groups.

intermediate

What does it mean if a logical expression is not in DNF?

It means the expression might have ORs inside ANDs or other complex nesting. DNF requires the top-level operator to be OR, with AND groups inside.

intermediate

Why might converting to DNF cause an expression to grow larger?

Because distributing AND over OR can duplicate parts of the expression, making it longer and more complex, especially with many variables.

Which of the following is a valid DNF expression?

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

B A AND (B OR C)

C (A AND B) OR (C AND NOT D)

D NOT (A OR B)

DNF requires OR of AND groups. Option C matches this. Option B has OR inside AND, not allowed in DNF.

In PHP, which operator represents logical AND?

A ||

B &&

C !

D ==

&& is the logical AND operator in PHP.

What does the 'NOT' operator do in logical expressions?

A Negates or reverses the truth value

B Combines two expressions with OR

C Combines two expressions with AND

D Checks equality

NOT reverses true to false and false to true.

Why is DNF useful in programming?

A It simplifies evaluation by standardizing structure

B It makes expressions harder to read

C It removes all negations

D It converts expressions to numbers

DNF standardizes logic to OR of ANDs, making evaluation easier.

What happens when you convert a complex expression to DNF?

A It always becomes shorter

B It removes all variables

C It becomes a single variable

D It may become longer due to distribution

Distributing OR over AND can duplicate parts, increasing length.

Explain what Disjunctive Normal Form (DNF) is and why it is useful in programming.

Think about how logic can be broken into simpler parts.

Describe how you would write a DNF logical expression in PHP code.

Remember PHP boolean operators.