Simplification Of Boolean Functions
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On first reading they simplify boolean expression rules almost impossible to solve. Larry has simplify boolean expression rules as many books as Fred. There are 75 books on the bookcase. How many books does Fred have? Then you realize how the problem can be redefined as an algebra problem. As an algebra problem the solution is much easier.
For many of the same reasons digital systems are based on an algebra--not the regular algebra you and I are familiar with but rather Boolean algebra. Boolean algebra is the theoretical foundation for digital systems. Boolean algebra formalizes the rules of logic. On the surface computers are great number crunchers, but inside computations are performed by binary digital circuits following the rules of logic.
We use Boolean algebra in this class to simplify Boolean expressions which represent circuits. In this lecture we will study algebraic techniques for simplifying expressions. In the next lecture we will look at mechanical ways--algorithms you can use with pencil and paper to simplify moderately complex Boolean functions and algorithms that machines can follow to simplify boolean expression rules arbitrarily complex Boolean functions.
There is a set B and two operators: There are at least two elements a and b in B such that a! It's simplify boolean expression rules necessary to provide a separate proof for the dual because of the principle of duality.
Any algebraic equality derived from the axioms of Boolean algebra remains true when simplify boolean expression rules operators OR and AND are interchanged and the identity elements 0 and 1 are interchanged.
This property is called the duality principle. Because of the duality principle, for any given simplify boolean expression rules we get it's dual for free. In this class we will use the axioms and theorems of Boolean algebra to simplify Boolean expressions. Using Boolean algebra to simplify Boolean expressions is an art. There is no algorithm you can follow that is guaranteed to result in the simplest form of the expression.
Like learning to play chess, with practice you will learn heuristics and begin to recognize patterns that will guide you to the solution. On the subject of simplification there is another more simplify boolean expression rules question that has to be answered, "What is simplification? An expression with the fewest operations? An expression with the fewest levels? The answer is it depends on what you are trying to optimize for.
Number of interconnections between gates? Before you can fully understand the relationship between these tradeoffs you need an understanding of how Boolean expressions are implemented with gates. This is the topic of the next section. A variable is a symbol that may take on the value 0 or 1. A literal is the use of a variable or its complement in an expression. A term is the expression formed by literals and operations at one level. For example, the following function: Has 3 variables x,y,z8 literals x,y,x,y',z,x',y,zand 4 terms xy, xy'z, x'yz, and the OR term that combines the first level AND terms.
Duality Any algebraic equality derived from the axioms of Boolean algebra remains true when the operators OR and AND are interchanged and the identity elements 0 and 1 are interchanged. The proof for DeMorgan's law using the axioms of Boolean algebra is long.
Another method that also works for the other theorems we just discussed is to show equality with a truth table: