Fundamental concepts of boolean algebra

1. Use of binary digits (Also visit Logic Gates)

In a normal algebraic expression, a variable can take any numerical value. For example, in the expression 3A+7B=C, we assume that A,B and C may range through the entire field of real numbers.

Since Boolean algebra deals with binary number system, the variables used in the boolean equations may assume only two possible values (0 and 1). If an equation describing logical circuitry has several variables, it is still understood that each of the variables can assume only the values 0 or 1. For example, in the equation A+B=C, each of the variables A,B and C may have only the values 0 or 1.

2. Logical addition

The symbol '+' is used for logical addition operator. It is also known as 'OR' operator. We can define the + symbol (OR operator) by listing all possible values of C in the equation A+B+C. It may be noted that since the variables A and B can have only two possible values (0 or 1) so only four (2 to the power of 2) combinations of inputs are possible. Consider this table

INPUTS OUTPUTS

A + B = C

0 -- 0 -- 0

0 -- 1 -- 1

1 -- 0 -- 1

1 -- 1 -- 1

The resulting output values for each of the four input combinations are given. The above table is called the truth table.

Observe that the result is 0 only when the value of both the input variables is 0. The result is 1 when any of the input variables is 1. Note that a result of 1 is also obtained when both the inputs A and B are 1. This is the reason why the + symbol does not have the 'normal' meaning, but is a logical additonal operator. This concept of logical addition may be executed to any number of variables.

3. Logical multiplication

The symbol '.' is used for logical multiplication operator. It is also known as 'AND' operator. We can again define the symbol (AND operator) by listing all possible combinations of A and B and the resulting value of C in the equation A.B=C. Consider the truth table for this logical operator as follows.

INPUTS OUTPUTS

A . B = C

0 -- 0 -- 0

0 -- 1 -- 0

1 -- 0 -- 0

1 -- 1 -- 1

The equation A.B=C is normally read as "A and B equals C".

4. Complementation

The two operations defined so far (OR and AND) are binary operations because they define an operation on two variables. The complementation operation is a unary operation which is defined on a single variable. The symbol '.' is normally used for complementation operator. It is also known as 'NOT' operator. Thus we write ~A, meaning take the complement of A, and complementation of a variable is the reverse of its value. If A=0 then ~A=1.

5. Operator precedence

Does A+B.C mean (A+B).C or A+(B.C)? The two generate different values for A-1, B=0 and C=0, we have (1+0).0=0 and 1+(0.0)=1, both are different. Hence it is necessary to define operator precedence in order to correctly evaluate Boolean expressions. The precedence of Boolean operators is as follows:

a. The expression is scanned from left to right

b. Expressions enclosed within parebtheses are evaliated first

c. All complement (NOT) operations are performed next

d. All '.' (AND) operations are performed ater that

e. Finally all '+' (OR) operations are performed in the end

So according to this precedence rule, A+B.C means A+(B.C).

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