1. TN 221: DIGITAL ELECTRONICS 1
Boolean algebra AND Basic logic gates

2. Boolean Algebra
In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of 1 or 0.
Operations in Boolean algebra include AND, OR and NOT.
Algebra is needed to express the output in terms of the input when construct a network of gates.

3. Boolean Algebra Boolean Addition
Addition is equivalent to the OR operation. The sum term is 1 if one or more of the literals are 1. The sum term is zero only if each literal is 0.
Boolean Multiplication
In Boolean algebra, multiplication is equivalent to the AND operation. The product of literals forms a product term. The product term will be 1 only if all of the literals are 1.

4. Boolean Algebra Boolean NOT (complementation)
Represents the inverse of a variable and is represented by overbar.
Thus

5. Boolean Algebra Operator Precedence
Each operator has a higher precedence level
The higher operator’s precedence level the earlier it is evaluated
Expression is scanned from left to right
First expressions enclosed within parentheses are evaluated
Then all complement (NOT) operations are performed
Then all AND operation are performed
Finally all OR operations are performed.

6. Rules and Laws of Boolean Algebra
Commutative Law
For addition, the commutative law states in terms of the result, the order in which variables are ORed makes no difference.
For multiplication, the commutative law states in terms of the result, the order in which variables are ANDed makes no difference.

7. Rules and Laws of Boolean Algebra
Associative law
For addition: The associative laws states that when ORing more than two variables, the result is the same regardless of the grouping of the variables.
For multiplication, the associative law states when ANDing more than two variables, the result is the same regardless of the grouping of the variables

8. Rules and Laws of Boolean Algebra
Distributive Law
This law states that ORing two or more variables and then ANDing the result with a single variable is equivalent to ANDing the single variable with each of the two or more variables and then ORing the products.
The distributive law is the factoring law. A common variable can be factored from an expression just as in ordinary algebra.

9. Rules and Laws of Boolean Algebra
Rules of Boolean algebra
Basic rules that are useful in manipulating and simplifying Boolean expressions.

10. Rules and Laws of Boolean Algebra
Duality
Given any logic expression, its dual is formed by replacing all + with ·, and vice versa and replacing all 0s with 1s and vice versa.

11. Rules of Boolean Algebra
Duality
Theorem
Its duality 1 2 3 4 5 6 7 8 9

12. Rules and Laws of Boolean Algebra
De Morgan’s Theorem
De Morgan’s 1st Theorem states that the complement of a product of variables is equal to the sum of the complemented variables
De Morgan’s 2nd Theorem states that The complement of a sum of variables is equal to the product of the complemented variables.

13. Boolean formulas and functions
A Boolean function is an expression formed by binary variables, binary operators( AND, OR and NOT), parentheses and equal sign.
The value of Boolean function can either be 0 or 1
A Boolean function may be represented as
An algebraic expression or
A truth table

14. Boolean formulas and functions
Representation as truth table x y z w 1

15. Boolean formulas and functions
Representation as truth table
The number of rows in the table is equal to 2n where n is the number of literals in the function.
The combinations of 0s and 1s for rows of this table are obtained from binary number counting from 0 to 2n-1

16. Boolean formulas and functions
These Boolean expressions can be used to describe or evaluate the output of a circuit.
Just like algebra, a set of rules exist that when applied to Boolean expressions can dramatically simplify them.
A simpler expression that produces the same output can be realized with fewer logic gates.
A lower gate count results in cheaper circuitry, smaller circuit boards, and lower power consumption.

17. Boolean formulas and functions
Using the rules and laws of Boolean algebra, a Boolean expression can be manipulated into whatever form best suits the needs of the computer system.
As far as the binary operation is concerned, two circuits are the same if their truth tables are equivalent.
The circuits, however, may not be the same when measuring performance or when counting the number of gates it took to implement the circuit.
The optimum circuit for a specific application can be designed using the tools presented in this chapter.

18. Boolean formulas and functions
Examples : Simplify the following Boolean expressions
𝑎𝑏+𝑐 𝑎𝑏+𝑑
𝑎+𝑏 𝑏 +𝑏
𝑏 𝑎+ 𝑎 𝑏
𝑎 +𝑏 𝑎+ 𝑏
𝑎 𝑏 𝑐 + 𝑎 𝑏 𝑐+ 𝑎 𝑏 𝑐 + 𝑎 𝑏𝑐