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Octal Another scheme, sometimes used in computer programming, is the octal number system, so named because it has eight symbols (according to our way of thinking), or 23. Every digit is an element of the set {0, 1, 2, 3, 4, 5, 6, 7}. This system is also known as base 8 or radix 8. Hexadecimal Another system used in computer work is the hexadecimal number system. It has 16 (24) symbols. These digits are the usual 0 through 9 plus six more, represented by A through F, the first six letters of the alphabet. The digit set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. This system is sometimes called base 16 or radix 16.
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Logic refers to the reasoning used by electronic machines. The term is also used in reference to the circuits that make up digital devices and systems.
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Boolean Algebra Boolean algebra is a system of mathematical logic using the numbers 0 and 1 with the operations AND (multiplication), OR (addition), and NOT (negation). Combinations of these operations are NAND (NOT AND) and NOR (NOT OR). This peculiar form of mathematical logic, which gets its name from the nineteenth-century British mathematician George Boole, is used in the design of digital logic circuits. Symbology In Boolean algebra, the AND operation, also called logical conjunction, is written using an asterisk (*), a multiplication symbol ( ), or by running two characters together, for example, X*Y. The NOT operation, also called logical inversion, is denoted by placing a tilde (~) over the quantity, as a minus sign ( ) or dash (-) followed by the quantity, as a lazy inverted L ( ) followed by the quantity, or as the quantity followed by an accent or prime sign ( ). An example is X. The OR operation, also called logical disjunction, is written using a plus sign (+), for example, X + Y. The foregoing are the symbols used by engineers. Table 26-1A shows the values of these functions, where 0 indicates falsity and 1 indicates truth. In mathematics and philosophy courses involving logic, you may see other symbols used for conjunction and disjunction. The AND operation in some texts is denoted by a detached arrowTable 26-1A.
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X 0 0 1 1 Y 0 1 0 1
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X 1 1 0 0 X*Y 0 0 0 1 X+Y 0 1 1 1
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Table 26-1B. Common theorems in Boolean algebra.
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Theorem (logic equation) X+0 = X X*1 = X X+1 = 1 X*0 = 0 X+X = X X*X = X ( X) = X X+( X) = 1 X*( X) = 0 X+Y = Y+X X*Y = Y*X X+(X*Y) = X X*( Y)+Y = X+Y X+Y+Z = (X+Y)+Z = X+(Y+Z) X*Y*Z = (X*Y)*Z = X*(Y*Z) X*(Y+Z) = (X*Y)+(X*Z) (X+Y) = ( X)*( Y) (X*Y) = ( X)+( Y) What it s called OR identity AND identity
Double negation Contradiction Commutativity of OR Commutativity of AND
Associativity of OR Associativity of AND Distributivity DeMorgan s Theorem DeMorgan s Theorem
head pointing up ( ) or by an ampersand (&), and the OR operation is denoted by a detached arrowhead pointing down ( ).
Theorems Table 26-1B shows several logic equations. Such facts are called theorems. Statements on either side of the equals sign in each case are logically equivalent. When two statements are logically equivalent, it means that one is true if and only if (iff ) the other is true. For example, the statement X = Y means that X implies Y, and also that Y implies X. Logical equivalence is sometimes symbolized by a double arrow with one or two shafts ( or ). Boolean theorems are used to analyze and simplify complicated logic functions. This makes it possible to build a circuit to perform a specific digital function, using the smallest possible number of logic switches.
Digital Circuits
All binary digital devices and systems employ high-speed electronic switches that perform Boolean operations. These switches are called logic gates. By combining logic gates, sophisticated digital systems can be built up. Even the most advanced computers are, at the basic level, comprised of logic gates.
Positive and Negative Logic Usually, the binary digit 1 stands for true and is represented by a voltage of about +5 V. The binary digit 0 stands for false and is represented by about 0 V. This is positive logic. There are