# Importance and Rules of Boolean Algebra Tutorial with Examples

## Boolean Algebra

Boolean algebra is an algebraic structure defined on a set of elements together with two binary operators (+) and (.)

### Closure

For and y in the alphabet A, x + y and x. y are also in A.

### Duality

If an expression  contains only the operations AND, OR and NOT. Then, the dual iof that expression is obtained by replacing each AND by OR, each OR by AND, all occurrences of 1 by 0 and all occurrences of 0 by 1.

### Key Points

Principle of duality is useful in determining the complement of a function.

## Boolean Operators

There are four Boolean operators

1. AND (•) operator (A .B)                2. OR (+) operator (A + B)
2. NOT (A / A’) operator
3. XOR (q operator (A+B=A-B+A•B)

### Operator Precedence

The operator for evaluating Boolean expression is

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

### Boolean Function

• A Boolean function is an expression formed with binary variables, the binary operators (+, .), the unary operator (—), parenthesis and equal sign.
• For a given value of variable, the function can take only one value either 0 or 1.
• A Boolean function can be shown by a truth table. To show a function in a truth table we need a list of the 2 combinations of ‘s and O’s of the n binary variables and a column showing the combinations for which the function is equal to 1 or 0. So, the table will have 2 rows and columns for each input variable and the final output.

## Canonical and Standard Form

• A canonical form is a unique representation for a given symbolic 1pression. If two expressions have the same canonical form, they must represent the same function.
• Boolean functions expressed as a sum of minterms i.e.. sum of products.
• A minterm is a special product of literals in which each input variable appears exactly once. Function with n variable has 2n
• A maxterm is represented as sum of variable. n variables can be combined to form 2n Each maxterm is obtained from OR of the nvariab►es, with each variable being unprimed, if the corresponding bit of the binary number is 0 and primed, if the binary number is 1.

Wherever value of function F1 is 1, corresponding minterm will be considered.

### Key Points

• Two functions of n binary variables are said to be equal, if they have same value for all possible 2′ combinations of the n variables.

## Representation of Boolean Function

### Using Minterm

A Boolean function may be expressed algebraically form given truth table by forming a minterm for each combination of the variable which produces a 1 In the function and taking the OR of all those terms. We can refer the Preceding table.

Thus, function F1 will be written as

F1 = xyz + xyz + xyz + xyz

Here, we have selected only those minterms for which

F1 = 1 and F1 will be represented (using minterm)

F1 = Ʃ (2, 4, 6, 7) = Z(m2, m4, m6, m7)

As, F1 = 1 for 010, 100, 110 and 111 those decimal equivalents are 2, 4, 6 and 7, respectively.

### Representation of Boolean Function using Maxterm

For maxterm representation use those maxterms for which F1 = 0 i.e., which produce 0 in the function. Then, form the AND of all those maxterm.

Thus, fl will be given by the below Boolean expression

= (x + y + z). (x + y+ z). (x +y +z). (x + y + z)

= Mo M1 M3 M5

F1 will be represented as

= П(0, 1, 3, 5)