DNF types (Disjunctive Normal Form) in PHP - Step-by-Step Execution

Concept Flow - DNF types (Disjunctive Normal Form)

Start with logical expression

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Convert to OR of ANDs form

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Identify each AND clause

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Combine clauses with OR

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Result: Expression in DNF

DNF means writing a logical expression as OR of AND groups. We break it down into AND parts, then join them with OR.

Execution Sample

PHP

<?php
$A = true;
$B = false;
$result = ($A && $B) || (!$A && $B);
echo $result ? 'true' : 'false';
?>

This code evaluates a DNF expression with variables A and B and prints true or false.

Execution Table

Step	Expression Part	Evaluation	Result
1	$A	true	true
2	$B	false	false
3	$A && $B	true AND false	false
4	!$A	NOT true	false
5	!$A && $B	false AND false	false
6	($A && $B) || (!$A && $B)	false OR false	false
7	echo $result	false	prints 'false'

💡 Expression fully evaluated, final result is false

Variable Tracker

Variable	Start	After Step 1	After Step 2	Final
$A	undefined	true	true	true
$B	undefined	undefined	false	false
$result	undefined	undefined	undefined	false

Key Moments - 2 Insights

Why is ($A && $B) false even though $A is true?

Because AND requires both sides true. $B is false (see step 2 and 3 in execution_table), so whole AND is false.

Why do we use OR between AND groups in DNF?

OR means if any AND group is true, whole expression is true. This matches DNF form (see step 6).

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of !$A at step 4?

A true

B false

C undefined

D error

💡 Hint

Check step 1 where $A is true, so NOT true is false at step 4.

At which step does the expression ($A && $B) || (!$A && $B) become false?

A Step 6

B Step 5

C Step 3

D Step 7

💡 Hint

Step 6 shows the OR of two false AND groups resulting in false.

If $B was true instead of false, what would be the result at step 6?

A undefined

B false

C true

D error

💡 Hint

If $B is true, then ($A && $B) is true AND true = true, so OR result is true.

Concept Snapshot

DNF means writing logic as OR of AND groups.
Syntax: (A && B) || (C && D) || ...
Each group joined by OR, inside groups use AND.
Evaluate AND groups first, then OR them.
Useful for simplifying or analyzing logic.

Full Transcript

This example shows how to evaluate a Disjunctive Normal Form (DNF) expression in PHP. We start with variables $A and $B. We check each part: $A is true, $B is false. Then we evaluate AND groups: $A && $B is false because $B is false. !$A is false because $A is true. So !$A && $B is false. Finally, we OR the two groups: false OR false is false. The program prints 'false'. This shows how DNF expressions are evaluated step by step.