**1. Introduction to Boolean Algebra**

- Boolean algebra is an algebraic structure defined on a set of elements together with two binary operators (+) and (.)
- A
**variable**is a symbol, for example, 'Α' used to represent a logical quantity, whose value can be**0**or**1**. - The
**complement of a variable**is the inverse of a variable and is represented by the variable over bar. - A
**literal**is a variable or the complement of a variable. **Boolean Value**- The value of Boolean variable can be either 1 or 0.

**Boolean Operators:**There are three basic Boolean operators- AND (∙) operator
- OR (+) operator
- NOT operator

### 2. Duality

- If an expression contains only the operations AND, OR and NOT. Then, the dual of that expression is obtained by replacing

each AND by OR,

each OR by AND,

all occurrences of 1 by 0,

all occurrences of 0 by 1.

The principle of duality is useful in determining the complement of a function.

- Logic expression: (
*x*•*y'*•*z*)*+*(*x*•*y*•*z'*)*+*(*y*•*z*) + 0 , - Duality of above logic expression is: (
*x*+*y'*+*z*) • (*x*+*y*+*z'*) • (*y*+*z*) • 1

**Boolean Function:**

- Any Boolean functions can be formed from binary variables and the Boolean operators.
- For a given value of the 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
^{n}combinations of 1's and 0'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^{n}rows and columns for each input variable and tile final output. - A function can be specified or represented in any of the following ways:
- A truth table
- A circuit
- A Boolean expression
- SOP (Sum Of Products)
- POS (Product of Sums)
- Canonical SOP
- Canonical POS

**Important Boolean operations over Boolean values:**

**3. Basic Theorems**

**Important Theorems used in Simplification**

**NOT-Operation theorem:**

**AND-Operation theorem:**

**OR-Operation theorem:**

**Distribution theorem:**

- A + BC = (A + B)(A + C)

**Others:**

- Consensus Theorem: This theorem is used to eliminate redundant term. It is applicable only when a boolean function contains three variables. Each variable used two times. Only one variable is complemented or uncomplemented. Then the related terms so that complemented or uncomplemented variable is the answer.

**4. MinTerm & MaxTerm**

**Minterm:**- Each product term is known as a minimum term that contains all the variables used in a function.
- A minterm is also called a
**canonical product term**. - A minterm is a product term, but a product term may or may not be a minterm.

**Maxterm:**- Each sum term is known as a maximum term that contains all of the variables used in the function.
- A
**maxterm**is a sum term of all variables in which each variable is either in complemented form or in uncomplemented form. - A maxterm is also called a
**canonical sum term**. - A maxterm is a sum term, but a sum term may or may not be a maxterm.

- The following are examples of product term, minterm, sum term, and maxterm for a function of three variables a, b, and c:
- product terms: a, ac, b’c, abc, a’bc, a’b’c’, …
- minterms: ab’c, abc, a’b’c, a’b’c’, …
- sum terms: a, (a+b), (b+c), (a’+b), (a’+b’), …
- maxterms: (a+b+c), (a+b’+c), (a’+b’+c’), …

**Representations of Minterm and Maxterm:**

**Note:**With'*n'*variables maximum possible minimum and maximum terms = 2^{n}- With'
*n'*variables maximum possible logic expression =

**5. SOP & POS**

**SOP****(Sum of Product):**A sum of product expression is two or more OR functions of AND functions.- SOP expression is used when output becomes logic 1.
- Example:

**POS****(Product of Sum):**It is the AND function of two or more OR function.- POS expression is used when output is logic '0'.
- Example:

**Example:**SOP and POS Equivalences for function F and Its Inverse F’.

**6. Duality Theorem**

- To convert positive logic into negative logic and
*vice-versa, a*dual function is used.- Change each AND sign by OR sign and
*vice versa*(↔ +) - Complement any 0 or l appearing in expression.
- Keep variable as it is.
- Example:

- Change each AND sign by OR sign and

**7. Minimization of Boolean Expressions**

The following two approaches can be used for simplification of a Boolean expression:

- Algebraic method (using Boolean algebra rules)
- Karnaugh map method

**Representation of K-map: **With n-variable Karnaugh-map, there are 2^{n} cells

**2 –variable K Map:**

**3 –variable K Map:**

**4 –variable K Map:**

**NOTE:**Once the Karnaugh map has been populated with**1**s,**0**s and**X**s as specified the only task that remains is to**group adjacent terms of the same state**(usually**1**) in groups of 2 raised to any rational power, i.e.**1, 2, 4, 8, 16, 32, 64**and so on. The larger the group the simpler the final expression. It is also possible for groups to overlap. This is often done to achieve a larger group size, hence simplifying the final expression.

**Minimization Procedure of Boolean Expression using K-map**

- Construct a K-map.
- Find all groups of horizontal or vertically adjacent cells that contain 1.
- Each group must be either rectangular or square with 1, 2, 4, 8, or 16 cells.
- Each group should be as large as possible.
- Each cell with 1 on the K-map must be covered at least once. The same
- the cell can be included in several groups if necessary.
- Select the least number of groups so as to cover all the 1’s.
- Adjacency applies to both vertical and horizontal borders.

- Translate each group into a product term. (Any variable whose value changes from cell to cell drops out from the term)
- Sum all the product terms.
**Note:**Don't care conditions can be used to provide further simplification of a Boolean Expression.

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