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In mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively. A logic function is an expression algebraically with binary variables, Logical operation symbols, parenthesis, and equal sign, which is known as a Boolean function. Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables. Boolean algebra traces its origins to an 1854 book by mathematician Geroge Boole. The distinguishing factor of Boolean algebra is that it deals only with the study of binary variables.
A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations ∧ (called “meet” or “and”), ∨(called “join” or “or”, a unary operation ¬ (called “complement” or “not”) and two elements 0 and 1 in A (called “bottom” and “top”, or “least” and “greatest” element, also denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following axioms hold.
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a ∨ (b ∨ c) = (a ∨ b) ∨ c a ∧ (b ∧ c) = (a ∧ b) ∧ c associativity a ∨ b = b ∨ a a ∧ b = b ∧ a commutativity a ∨ (a ∧ b) = a a ∧ (a ∨ b) = a absorption a ∨ 0 = a a ∧ 1 = a identity a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) distributivity a ∨ ¬a = 1 a ∧ ¬a = 0 complements
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Basic operations:
The basic operations of Boolean algebra are as follows:
- AND (conjunction), denoted x∧y (sometimes x AND y or Kxy), satisfies x∧y = 1 if x=y=1, and x∧y = 0 otherwise.
- OR(disjunction), denoted x∨y (sometimes x OR y or Axy), satisfies x∨y = 0 if x = y = 0, and xVy = 1 otherwise.
- NOT(Negation), denoted Ā(sometimes NOT A, NA or !A), satifies Ā = 0 if A = 1 and Ā = 1 if A = 0
Logic gates:
1. AND Gate:
The AND Gate contains two or more input values that produce only one output value. AND gate produces 1 output when all inputs are 1, otherwise, the output will be 0. It can be explained with the help of two switches connected in series. In the AND gate, the current is flowing in the circuit only when both switches, A, and B, are closed. The switch contains two states which are ON or OFF. The ON means logic 1 and the OFF means logic 0. So, when both switches are ON, the output is 1 and when any of the switches are OFF, the output is 0. There are three symbols for AND gates: the American (ANSI or ‘military’) symbol and the IEC (‘European’ or ‘rectangular’) symbol, as well as the deprecated DIN symbol. Additional inputs can be added as needed. For more information see Logic Gate Symbol.s It can also be denoted as the symbol “^” or“&”.The gate with inputs A and B and output C implements the logical expression {\displaystyle C=A\cdot B}. This expression also may be denoted as C=A^B or C=A&B.
The graphical symbol, a logical circuit, algebraic expression, and truth table of the AND gate is shown below:
2. NAND gate:
The NAND Gate contains two or more two input values that produce only one output value. This gate is the combination of AND and NOT gates. This gate is a complement of AND function. This gate produces output 0 when all inputs are 1, otherwise, the output will be 1. There are three symbols for NAND gates: the MIL/ANSI symbol, the IEC symbol, and the deprecated DIN symbol sometimes found on old schematics. For more information see logic gate symbols. The ANSI symbol for the NAND gate is a standard AND gate with an inversion bubble connected.
The graphical symbol, algebraic expression, and truth table of the NAND gate are shown below:
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