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The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
Boolean expressions are represented very similarly to algebraic expressions, the only difference being the specific values and operators used. Boolean expressions use true and false as constant values, and the operators include ∧ {\displaystyle \land } ( AND ), ∨ {\displaystyle \lor } ( OR ), ¬ {\displaystyle \neg } ( NOT ).
A complete table of "logic operators" is shown by a truth table, giving definitions of all the possible (16) truth functions of 2 boolean variables (p, q): p q
In Boolean algebra, a monotonic function is one such that for all a i and b i in {0,1}, if a 1 ≤ b 1, a 2 ≤ b 2, ..., a n ≤ b n (i.e. the Cartesian product {0, 1} n is ordered coordinatewise), then f(a 1, ..., a n) ≤ f(b 1, ..., b n). In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of ...
Boolean algebra (structure) Boolean function; Digital circuit; Four-valued logic; Homogeneity (linguistics) Paraconsistent logic § An ideal three-valued paraconsistent logic; Setun – an experimental Russian computer which was based on ternary logic; Ternary numeral system (and Balanced ternary) Three-state logic (tri-state buffer) The World ...
The material conditional (also known as material implication) is an operation commonly used in logic.When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false.
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected ...