# 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.

**Dualit****y**

If an expression contains only the operations AND, OR and NOT. Then, the dual _{i}of 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*

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

**Operator Precedence**

The operator for evaluating Boolean expression is

**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.

**C****a****nonical 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_{r}epresent 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 2^{n} - A maxterm is represented as sum of variable.
*n*variables can be combined to form 2^{n}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 F_{1} 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. -
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**Representation of Boolean Function**

**Using Minterm**

A Boolean function may be expressed algebraically form given truth table by ^{f}orming 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 F_{1} will be written as

**F1 = xyz + xyz + xyz + xyz**

Here, we have selected only those minterms for which

F_{1} = 1 and F_{1} will be represented (using minterm)

**F _{1} = Ʃ (2, 4, 6, 7) = Z(m_{2}, m_{4}, m_{6}, m_{7})**

As, F_{1} = 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 F_{1} = 0 *i.e., *which produce 0 in the function. Then, form the AND of all those maxterm.

Thus, *f _{l}* will be given by the below Boolean expression

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

**= M _{o} M_{1} M_{3} M_{5}**

F_{1} will be represented as

= **П(0, 1, 3, 5)**

# Sorting in Design and Analysis of Algorithm Study Notes with Example

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