Information AboutTruth Tables |
| CATEGORIES ABOUT TRUTH TABLE | |
| boolean algebra | |
| mathematical tables | |
|
(Expressions may be Arguments ; i.e., a Conjunction of expressions, each Conjunct of which is a Premise with the last being the Conclusion .) "The pattern of reasoning that the truth table tabulates was '' uses them to place Truth Function s in a series. The wide influence of this work led to the spread of the use of truth tables. Truth tables are used to compute the values of propositional expressions in an effective manner that is sometimes referred to as a '' Decision Procedure ''. A propositional expression is either an Atomic Formula — a propositional constant, propositional variable, or propositional function term (for example, ''Px'') — or built up from atomic formulas by means of logical operators, for example, AND (), OR (), NOT (). For instance, is a propositional expression. The column headings on a truth table show (i) the propositional functions and/or variables, and (ii) the truth-functional expression built up from those propositional functions or variables and operators. The rows show each possible valuation of T or '''F''' assignments to (i) and (ii). In other words, each row is a distinct interpretation of (i) and (ii). Truth tables for classical (i.e., bivalent) logic are limited to Boolean Logic systems where only two Truth Value s are possible, ''true'' or ''false'', usually denoted simply T and '''F''' in the tables (as remarked above). LOGICAL CONJUNCTION For example, take two propositional variables, and , and the logical operator " AND " (∧), signifying the conjunction "A and B" or ∧ . In common English, if both A and B are true, then the conjunction " ∧ " is true; under all other possible assignments of truth values to ∧ , the conjunction is false. This relationship is defined as follows: LOGICAL DISJUNCTION The OR (∨) relationship is defined as follows: NEGATION In a boolean logic system, all the operators can be explicitly defined this way. For example, the NOT (¬) relationship is defined as follows: LOGICAL NAND Compound expressions can be constructed, using parenthesis to denote precedence.
Key: :T = true, F = false :∧ = AND (logical conjunction) :∨ = OR (logical disjunction) :∧ = XOR (exclusive or) :∨ = XNOR (exclusive nor) :→ = Conditional "if-then" :← = Conditional "(then)-if" <↔>: (exclusive nor). Johnston Diagram s, similar to Venn Diagram s and Euler Diagrams , provide a way of visualizing truth tables. An interactive Johnston Diagram illustrating truth tables is at LogicTutorial.com CONDENSED TRUTH TABLES FOR BINARY OPERATORS For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example Boolean Logic uses this condensed truth table notation:
|