Is 1 true or false? 1 = TRUE, 0 = FALSE in Boolean logic
In Boolean logic, 1 means TRUE and 0 means FALSE: they are the only two values, written TRUE and FALSE in the words and programming notations and 1 and 0 in the mathematical and algebra notations. A constant in a formula either disappears or decides the result, and the laws of identity and domination say which.
The first table below shows A AND TRUE, whose column is simply A: TRUE is the neutral element of AND, as FALSE is of OR (A OR FALSE is A). The second shows A OR TRUE, which is TRUE on both rows: TRUE dominates OR, as FALSE dominates AND (A AND FALSE is FALSE). The simplifier applies these two laws whenever a complement law has produced a constant, which is how A AND (B OR NOT B) collapses to A.
The parser accepts TRUE, FALSE, 1 and 0, case-insensitively for the words, and prints the constants to match the notation: words and programming notation print TRUE and FALSE, the mathematical and algebra notations print 1 and 0. The digits are Boolean constants, not integers: there is no arithmetic, 1 AND 1 is 1, and A = 1 is not a formula, because = is not an operator in this language.
A formula made only of constants has no inputs and therefore a truth table with exactly one row, which the semantics specification fixes so that checks and exports agree on the edge case; NOT TRUE is FALSE by the constant-negation law. In code, Python writes True and False, SQL has TRUE, FALSE and the third value NULL, and C treats zero as false and any other number as true, three conventions the languages reference compares.
Accepted spellings
TRUE FALSE 1 0
TRUE / FALSETRUE / FALSE1 / 01 / 0A AND TRUE, A OR FALSE: A AND TRUE
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | TRUE |
Beginnerf is true exactly when both A and TRUE are true.
In every other case, f is false.
About the operators:
- AND is true only when all of its parts are true.
- TRUE and FALSE are constants: they do not depend on any input.
Intermediatef is true exactly when both A and TRUE are true.
In every other case, f is false.
Advancedf is true exactly when both A and TRUE are true [conjunction ∧].
In every other case, f is false.
Formal notation: A ∧ 1
A OR TRUE, A AND FALSE: A OR TRUE
| # | A | f |
|---|---|---|
| 0 | FALSE | TRUE |
| 1 | TRUE | TRUE |
Beginnerf is true exactly when at least one of A or TRUE is true (both may be true).
In every other case, f is false.
About the operators:
- OR is inclusive: it is true when at least one part is true, including when several are.
- TRUE and FALSE are constants: they do not depend on any input.
Intermediatef is true exactly when at least one of A or TRUE is true (both may be true).
In every other case, f is false.
Advancedf is true exactly when at least one of A or TRUE is true (both may be true) [inclusive disjunction ∨].
In every other case, f is false.
Formal notation: A ∨ 1
The common slip
Writing a constant where a variable was meant, or the reverse. A rule that reads member AND TRUE has a redundant condition, and the simplifier removes it; a rule that reads member AND FALSE can never be true, and the contradiction check says so on every row.
Questions
Is 0 ever TRUE?
Not in Boolean logic: 0 is FALSE and 1 is TRUE. The confusion comes from conventions outside logic. In a Unix shell an exit status of 0 means success, so if takes its first branch when a command returns 0; that is a convention about programs, not a truth value. In C, Python and JavaScript the number 0 counts as false and other numbers such as 1 and -1 count as true, so bool(-1) is True in Python.
Does 1 AND 1 equal 2?
No. 1 is TRUE, 1 AND 1 is TRUE, and there is no addition; the algebra plus sign A + B means OR, so 1 + 1 is 1.
How do I say that A is TRUE?
Just A. A formula is a claim about the rows where it is TRUE, so the formula A holds exactly on the rows where A is TRUE; A = TRUE is rejected by the parser, and A <-> TRUE simplifies to A.
Related
Guides
Laws
- Identity law of Boolean algebra: A AND TRUE = A, A OR FALSE = A
- Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE
- Complement law: A AND NOT A = FALSE, A OR NOT A = TRUE
- Negating constants: NOT TRUE = FALSE, NOT FALSE = TRUE
- XOR identities: A XOR FALSE = A, A XOR TRUE = NOT A, A XOR A = FALSE
- Huntington's postulates of Boolean algebra, with their duals
Tools
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
- Logic circuit generator: Boolean expression to gate diagram (SVG)
See also
- Identity law of Boolean algebra: A AND TRUE = A, A OR FALSE = A
- Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE
- Negating constants: NOT TRUE = FALSE, NOT FALSE = TRUE
- Complement law: A AND NOT A = FALSE, A OR NOT A = TRUE
- “=” is not a Boolean operator: use ↔ or compare the formulas
- NULL is not FALSE: three-valued logic in SQL
- Boolean logic in SQL: AND, OR, NOT and the NULL three-valued trap
- Boolean logic in Python: and, or, not, precedence and common mistakes
Operators and gates
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