Boolean Algebra Operator Precedence

A thumbnail of the Boolean Precedence Visualizer

Boolean Precedence Visualizer

Enter a Boolean expression using T (true), F (false), ! (NOT), && (AND), || (OR), and (). Click 'Evaluate' to see the order of operations.

Evaluation steps will appear here.

Operator precedence in Boolean algebra

Operator precedence (also called operator order) tells you which Boolean operation is evaluated first when an expression contains several operators and no parentheses. Using precedence correctly avoids ambiguity and helps you evaluate or simplify expressions step by step.

Common precedence order (highest → lowest)

Priority Operator(s) Typical symbol(s) Description
1 (highest) Parentheses / Grouping ( ... ) Force evaluation of the grouped subexpression first
2 NOT (negation) ¬, !, ~ Unary negation — inverts a value
3 AND (conjunction) ∧, &, · True only if both operands are true
4 OR (disjunction) ∨, |, + True if at least one operand is true
5 XOR (exclusive OR) ⊕ True when operands differ
6 (lowest) Implication / Equivalence → (implies), ↔ (equivalence) Logical implication and bi-conditional — usually lowest precedence

Short, practical rules

  • Always evaluate anything inside parentheses first.
  • NOT applies to the smallest possible operand to its right (or the grouped operand immediately following it).
  • AND binds tighter than OR — e.g. A ∨ B ∧ C means A ∨ (B ∧ C).
  • If you are unsure which operators a particular programming language or tool uses, use parentheses to make your intention explicit — it avoids mistakes.

Examples (step-by-step)

Example 1: A ∨ B ∧ ¬C
Step 1: NOT first → ¬C
Step 2: AND next → B ∧ (¬C)
Step 3: OR last → A ∨ (B ∧ ¬C)
Example 2: ¬A ∧ (B ∨ C) → D
Step 1: Parentheses first → B ∨ C
Step 2: NOT → ¬A
Step 3: AND → (¬A) ∧ (B ∨ C)
Step 4: Implication last → ((¬A) ∧ (B ∨ C)) → D

Truth table reminders (basic operators)

A B A ∧ B A ∨ B A ⊕ B ¬A
0 0 0 0 0 1
0 1 0 1 1 1
1 0 0 1 1 0
1 1 1 1 0 0

Notes & best practices

• Operator symbols and exact precedence can vary between textbooks, logic systems, and programming languages. For example, some languages implement bitwise operators with different precedence rules or include additional operators (NAND, NOR, etc.).
• When writing complex expressions, prefer using parentheses to document your intention clearly — they make expressions unambiguous and easier to debug.
• In digital logic design, the same precedence applies conceptually, but designers usually show circuits or use parentheses (or draw the logic explicitly), so ambiguity is avoided.

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