Boolean Math and Models
The Core Idea
Boolean algebra describes decisions whose results are TRUE or FALSE. A Boolean model writes a control rule as an expression, so you can reason about that rule before choosing how to draw or program it.
What You Will Learn
Boolean algebra notation and the three core operators (AND, OR, NOT)
How to read and write Boolean expressions for control logic
De Morgan's theorems and how they simplify ladder logic
Truth tables as a verification tool
How Boolean expressions map directly to ladder rungs
Why This Matters
Boolean algebra is the mathematical language behind every ladder rung. When you can read a Boolean expression, you can predict exactly what a rung will do without tracing power flow. When you can write one, you can design logic on paper before ever opening a programming environment.
Notation
| Operation | Symbol | Meaning | Ladder Equivalent |
| AND | · or ∧ | Both must be true | Series contacts |
| OR | + or ∨ | Either is sufficient | Parallel branches |
| NOT | ¬ or overbar | Inversion | Normally closed (NC) contact |
Examples:
Y = A · B→ Y is true when A AND B are both trueY = A + B→ Y is true when A OR B (or both) are trueY = ¬A→ Y is true when A is false
Operator Precedence
Just like arithmetic (multiply before add), Boolean has precedence:
NOT (highest — evaluated first)
AND (middle)
OR (lowest — evaluated last)
So A + B · ¬C means A + (B · (¬C)) — NOT C first, then AND with B, then OR with A.
Use parentheses to override: (A + B) · ¬C means OR first, then AND with NOT C.
De Morgan's Theorems
These two rules let you push NOT through AND/OR:
¬(A · B) = ¬A + ¬B (NOT of AND = OR of NOTs)
¬(A + B) = ¬A · ¬B (NOT of OR = AND of NOTs)Why this matters for ladder: If you need "NOT (A AND B)", you can implement it as two NC contacts in parallel instead of trying to negate an entire series path.
Building a Boolean Model
For any control requirement:
Identify the output — what tag should be controlled?
List conditions — what inputs affect it?
Classify each condition:
Must be TRUE → variable (e.g.,
SENSOR)Must be FALSE → negated variable (e.g.,
¬FAULT)
Determine relationships:
ALL required → AND (·)
ANY sufficient → OR (+)
Write the expression
Example
Requirement: "Run the pump if level is high AND valve is open, OR if manual override is active."
Model:
PUMP = (LEVEL_HIGH · VALVE_OPEN) + MANUAL_OVRVerifying with Truth Tables
For any Boolean expression, build the truth table by evaluating every input combination:
| LEVEL_HIGH | VALVE_OPEN | MANUAL_OVR | PUMP |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
If the truth table matches your requirement for every row, your expression is correct.
From Boolean to Ladder
The translation is mechanical:
Each AND (·) → series contacts in the same branch
Each OR (+) → parallel branches
Each NOT (¬) → NC contact instead of normally open (NO) contact
Parentheses → define branch grouping
PUMP = (LEVEL_HIGH · VALVE_OPEN) + MANUAL_OVRBecomes: two parallel branches — top has NO LEVEL_HIGH and NO VALVE_OPEN in series, bottom has NO MANUAL_OVR alone — both feeding the PUMP output coil.
Common Boolean Identities
| Identity | Rule |
| A · 1 = A | AND with TRUE has no effect |
| A · 0 = 0 | AND with FALSE always gives FALSE |
| A + 0 = A | OR with FALSE has no effect |
| A + 1 = 1 | OR with TRUE always gives TRUE |
| A · A = A | Idempotent AND |
| A + A = A | Idempotent OR |
| A · ¬A = 0 | Contradiction |
| A + ¬A = 1 | Tautology |
These help simplify complex expressions before translating to ladder.
Key Takeaways
Boolean algebra has three operators: AND (·), OR (+), NOT (¬)
Precedence: NOT > AND > OR (use parentheses to clarify)
De Morgan's theorems let you push NOT through AND/OR
Always build the Boolean model BEFORE drawing ladder
Truth tables verify correctness for every input combination
The translation from Boolean to ladder is mechanical and reversible
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