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7: Boolean Algebra

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  • All arithmetic operations performed with Boolean quantities have but one of two possible outcomes: either 1 or 0. There is no such thing as “2” or “-1” or “1/2” in the Boolean world. It is a world in which all other possibilities are invalid by fiat. As one might guess, this is not the kind of math you want to use when balancing a checkbook or calculating current through a resistor. However, Claude Shannon of MIT fame recognized how Boolean algebra could be applied to on-and-off circuits, where all signals are characterized as either “high” (1) or “low” (0). 

    • 7.1: Introduction to Boolean Algebra
    • 7.2: Boolean Arithmetic
    • 7.3: Boolean Algebraic Identities
      In mathematics, an identity is a statement true for all possible values of its variable or variables. The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original “anything,” no matter what value that “anything” (x) may be. Like ordinary algebra, Boolean algebra has its own unique identities based on the bivalent states of Boolean variables.
    • 7.4: Boolean Algebraic Properties
    • 7.5: Boolean Rules for Simplification
      Boolean algebra finds its most practical use in the simplification of logic circuits. If we translate a logic circuit’s function into symbolic (Boolean) form, and apply certain algebraic rules to the resulting equation to reduce the number of terms and/or arithmetic operations, the simplified equation may be translated back into circuit form for a logic circuit performing the same function with fewer components.
    • 7.6: Circuit Simplification Examples
    • 7.7: The Exclusive-OR Function - The XOR Gate
      One element conspicuously missing from the set of Boolean operations is that of Exclusive-OR, often represented as XOR. Whereas the OR function is equivalent to Boolean addition, the AND function to Boolean multiplication, and the NOT function (inverter) to Boolean complementation, there is no direct Boolean equivalent for Exclusive-OR. This hasn’t stopped people from developing a symbol to represent this logic gate, though.
    • 7.8: DeMorgan’s Theorems
      A mathematician named DeMorgan developed a pair of important rules regarding group complementation in Boolean algebra. By group complementation, I’m referring to the complement of a group of terms, represented by a long bar over more than one variable.
    • 7.9: Converting Truth Tables into Boolean Expressions
      In designing digital circuits, the designer often begins with a truth table describing what the circuit should do. The design task is largely to determine what type of circuit will perform the function described in the truth table. There are procedural techniques available and Boolean algebra proves its utility in a most dramatic way.