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Boolean Expression Evaluator – Evaluate Logic Expressions Online

Evaluate any Boolean expression for given variable values with our free online Boolean expression evaluator. Supports all logical operators including AND, OR, NOT, XOR, NAND, and NOR.

A =TRUE
B =FALSE

Use variables: A, B

Examples:

Boolean Expression Evaluator – Logic Calculator

Boolean algebra is the mathematics of logic. Developed by George Boole in 1854, it uses variables that can only be TRUE or FALSE, combined with logical operators. This system forms the foundation of all digital circuits – every processor, memory chip, and logic gate operates on Boolean principles.

This evaluator lets you test logical expressions with custom variable values. Whether you're designing digital circuits, writing conditional statements in code, or studying discrete mathematics, you can verify your logic expressions instantly.

Logical Operators

AND

TRUE AND TRUE = TRUE
Otherwise = FALSE

Both conditions must be true. Like a series circuit – current flows only if both switches are closed.

OR

FALSE OR FALSE = FALSE
Otherwise = TRUE

At least one condition must be true. Like a parallel circuit – current flows if either switch is closed.

NOT

NOT TRUE = FALSE
NOT FALSE = TRUE

Inverts the value. Also called negation or complement. Flips TRUE to FALSE and vice versa.

XOR (Exclusive OR)

Same values = FALSE
Different values = TRUE

True when exactly one input is true. Used in adders, comparators, and encryption.

NAND (NOT AND)

TRUE NAND TRUE = FALSE
Otherwise = TRUE

The opposite of AND. NAND gates are "functionally complete" – you can build any logic circuit using only NAND gates.

NOR (NOT OR)

FALSE NOR FALSE = TRUE
Otherwise = FALSE

The opposite of OR. Like NAND, NOR gates are functionally complete – any circuit can be built from NOR alone.

Truth Tables

AND Truth Table

A | B | A AND B
------|-------|--------
FALSE | FALSE | FALSE
FALSE | TRUE | FALSE
TRUE | FALSE | FALSE
TRUE | TRUE | TRUE

OR Truth Table

A | B | A OR B
------|-------|-------
FALSE | FALSE | FALSE
FALSE | TRUE | TRUE
TRUE | FALSE | TRUE
TRUE | TRUE | TRUE

XOR Truth Table

A | B | A XOR B
------|-------|--------
FALSE | FALSE | FALSE
FALSE | TRUE | TRUE
TRUE | FALSE | TRUE
TRUE | TRUE | FALSE

Worked Examples

Example 1: Simple AND

Evaluate: A AND B, where A = TRUE, B = FALSE

Substitute: TRUE AND FALSE
AND requires both to be TRUE
Result: FALSE

Example 2: Compound Expression

Evaluate: (A AND B) OR (NOT C), where A = TRUE, B = FALSE, C = TRUE

Step 1: Evaluate A AND B
TRUE AND FALSE = FALSE
Step 2: Evaluate NOT C
NOT TRUE = FALSE
Step 3: Evaluate FALSE OR FALSE
Result: FALSE

Example 3: XOR Logic

Evaluate: A XOR B, where A = TRUE, B = FALSE

XOR is TRUE when inputs differ
TRUE XOR FALSE = TRUE
(One is TRUE, one is FALSE – they differ)
Result: TRUE

Example 4: NAND Gate

Evaluate: A NAND B, where A = TRUE, B = TRUE

NAND = NOT (A AND B)
A AND B = TRUE AND TRUE = TRUE
NOT TRUE = FALSE
Result: FALSE

Example 5: Complex Expression

Evaluate: (A OR B) AND (NOT A OR C), where A = FALSE, B = TRUE, C = FALSE

Step 1: A OR B = FALSE OR TRUE = TRUE
Step 2: NOT A = NOT FALSE = TRUE
Step 3: NOT A OR C = TRUE OR FALSE = TRUE
Step 4: TRUE AND TRUE = TRUE
Result: TRUE

Example 6: NOR Gate

Evaluate: A NOR B, where A = FALSE, B = FALSE

NOR = NOT (A OR B)
A OR B = FALSE OR FALSE = FALSE
NOT FALSE = TRUE
Result: TRUE

Quick Fact

George Boole published "The Laws of Thought" in 1854, creating Boolean algebra. He died in 1864, long before his work became the foundation of digital computing. In 1937, Claude Shannon's master's thesis showed how Boolean algebra could optimize relay circuits – launching the digital age.

Frequently Asked Questions

What is Boolean algebra used for?

Boolean algebra is the mathematical foundation of all digital logic. It's used in circuit design, programming conditionals, database queries, search engines, and anywhere decisions depend on multiple conditions. Every if-statement in code uses Boolean logic.

What's the order of operations for Boolean expressions?

NOT has highest precedence, then AND, then OR. XOR, NAND, and NOR have varying precedence depending on context. Use parentheses to make your intended order explicit: NOT A AND B means (NOT A) AND B, not NOT (A AND B).

How do I simplify Boolean expressions?

Use Boolean algebra laws: De Morgan's laws (NOT (A AND B) = NOT A OR NOT B), distributive law (A AND (B OR C) = (A AND B) OR (A AND C)), and absorption (A OR (A AND B) = A). Karnaugh maps help visualize simplifications for complex expressions.

What's the difference between XOR and OR?

OR is TRUE if at least one input is TRUE (including both). XOR is TRUE only if exactly one input is TRUE – not both. XOR means "one or the other, but not both." OR means "one or the other, or both."

Can I use numbers instead of TRUE/FALSE?

In many programming languages, 0 is FALSE and any non-zero value is TRUE. This calculator uses explicit TRUE/FALSE for clarity. In Boolean algebra, the values are often written as 1 (TRUE) and 0 (FALSE).

What are De Morgan's Laws?

De Morgan's Laws show how to distribute NOT over AND/OR: NOT (A AND B) = NOT A OR NOT B, and NOT (A OR B) = NOT A AND NOT B. These are essential for simplifying expressions and designing circuits with only NAND or only NOR gates.

How many possible Boolean functions exist for n variables?

For n variables, there are 2^n possible input combinations, and each can map to TRUE or FALSE. So there are 2^(2^n) possible Boolean functions. For 2 variables: 2^(2^2) = 16 functions. For 3 variables: 2^(2^3) = 256 functions.

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