Boolean algebra (or less commonly symbolic logic) is a branch algebra that deals with only two logic values - 0 (corresponding to false) and 1 (corresponding to true).
Today, Boolean algebra is the primary mathematical tool used in designing modern digital systems. Switching functions are described using Boolean algebra since they deal with two discrete states - ON and OFF (or 1 and 0). Those functions are in turn implemented via transistors which act as switches, a natural implementation for representing Boolean algebra operations. Once primitive Boolean operation circuits such as NOT, AND, and ORgates are implemented, any conceivable system of logic can be implemented using them like Lego pieces.
Boolean algebra uses variables just like normal algebra. Those variables can only have one of two values - either a 0 or a 1. Variable are commonly represented as a single alphabet letter. While there is no one acceptable convention, a it's not uncommon to see letters such as used for inputs and for output. That's also the convention used on WikiChip. Sometimes it's desired to represent the negated (opposite) value of a variable, that's often done with a bar or a tick (prime) above or next to the letter, for example or although other notations exist. is read "not A", regardless of notation.
Boolean algebra has a set of operations that can be performed on Boolean values, those operations are conveniently enough called binary operations. The three common Boolean operators are AND, OR, and NOT. Understanding those operators can better be done by examining their behavior via tool called a truth table. truth tables is a table that lists all possible input values and their respective output values. Truth tables are an excellent way of seeing the relationships between input values and given Boolean expressions.
The Boolean operator AND is usually represented by either , , or no symbol at all: for example "", "", and "" are all equivalent and are read "A AND B". The behavior of this operator is shown in the truth table on the right. The result of "A AND B" is true if both A and B are true; otherwise the result is false. This expression is also called a Boolean product.
The Boolean operator OR is usually represented by or operators. For example "" and "". The expression is read "A or B". The result of "A OR B" is true if either A is true or B is true; otherwise the result is false. This expression is also called a Boolean sum.
The Boolean operator NOT is represented by many notations, the three most popular ones are "", "", and "". Note that unlike the AND and OR operators, the NOT operator is a unary operator and is thus drawn above or on the side of the variable. The expression is read "not A". The truth table for the NOT operator is shown on the right. The result of the NOT operator is true if A is false, otherwise the result is true. This expression is called a Boolean complement.
So far we've made it simple by explicitly using parenthesis in all of our examples to indicate a certain part of the expression is evaluated before another part. The order of operations of a Boolean expression is very important to obtain correct result. For example consider the function for input . Does it mean ? or does it mean ? Same expression, different results. It turns out the the correct order is (and ). In Boolean expressions, the NOT operator has the highest precedence, followed by AND, then OR.
Boolean algebra is govern by a set of special axioms that say what kind of Boolean expression manipulations can be done. They are called axioms because they not things that have to be proven but rather part of the definition of Boolean algebra. Many of those laws are common to both Boolean algebra and ordinary algebra. Using those laws, equations can be converted into different forms. One particular transformation known as minimization plays a crucial role in the design of logic circuits. One last thing to note before we get to the actual laws is that Boolean algebra identities come in pairs. This is known as duality principle and it is covered in much more detail later on.
It's interesting to note that it's easy to see the divergence between Boolean algebra and ordinary algebra from those laws. For example consider . From Dominance Law we know the answer is . This is clearly not true for ordinary algebra where . Likewise from the Absorption Law we know that while in ordinary algebra this is not true either.
Axioms explanation
The Identity Axiom simply states that any expression ANDed with 1 or ORed with 0 results in the original expression. Identity elements or simply identities are elements that when used with their appropriate operator leave the original element unchanged. In the case of Boolean algebra, the identity element for AND is 1 and 0 for OR.
The Commutative Axiom states that individual elements in an expressions can be reordered without affecting the meaning of the expression. For example .
The Associative Axiom states that individual elements in an expression can be regrouped without affecting the meaning of the expression. For example . Simply put, it makes no difference in what order you group the expressions when ANDing or ORing several expressions together.
The Distributive Axiom states that ANDing distributes over over ORing. That is, ORing several expressions and ANDing the result is equaivilent to ANDing the result with a each of the individual expressions then ORing the product. Often times the Distributive Axiom is applied in reverse in a similar way to factoring in ordinary algebra. For example, given , the expression can be factored out to
Minimization
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Complementary Function
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Canonical Form
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