Logic Operations - Digital Logic Design

Binary logic has two binary operators denoted by `+` and `-` Operators.

The basic logic operations are AND, OR and NOT. Boolean expression consists of logic operators and logic variables for example x + xy is a Boolean expression with `2` logic variables x and y and with one Boolean operator `+`.

The operator precedence for evaluating Boolean expression is

1. parentheses
2. NOT
3. AND
4. OR.

Truth table for AND operation

A	B	Y
0	0	0
0	1	0
1	0	0
1	1	1

Let A and B be the two Boolean variables, performing an AND operation on these variables leads to a result Y.

AND operation results in True or 1 when the input variables are true or 1 otherwise false or 0. AND operation is referred to as logical multiplication. Because it leads to a product of variables, the dot symbol is used to represent AND operation. i.e A.B

Truth table for OR operation.

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	1

The symbol for OR operation is `+`. It is also referred to as logical sum. This operation is sometimes called as inclusive – OR.

When any one of the variable is true the OR operation results in true.

Truth table for NOT operation

A	Y
0	1
1	0

It is denoted by A1 or Ā this is similar to the complement operation.

Dual and complement Boolean expression: The term literal refers to both the variables and their complements.

The complement of a logical expression is obtained by

1. Changing all `•` to `+` 
2. Changing all `+` to `•`
3. Changing all 1` to 0`
4. Changing all 0` to 1` and
5. Complementing each literal.

For example: Obtain the complement of 1.A + BC + 0
complement is (0 + A) (B + C).1

The dual of a Boolean expression is obtained by

1. Changing all `•` to `+`
2. Changing all `+` to `•` 
3. Changing all 1` to 0`
4. Changing all 0` to 1` and
5. But not complementing any literal.

Example

Que.     Obtain the dual of 1.A + BC + 0

Ans.  (0 + A) (B+ C) 1 is the dual of the given boolean expression.

Special forms of boolean expressions

There are 4 forms of boolean expressions.

1. Sum of products
2. Product of sums
3. Canonical sum of products
4. Canonical product of sums