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# Boolean Expression.pptx

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# Boolean Expression.pptx

Boolean algebra

Boolean algebra

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### Boolean Expression.pptx

1. 1. Boolean Algebra
2. 2. Boolean Algebra • Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables. • Boolean algebra traces its origins to an 1854 book by mathematician George Boole.
3. 3. Boolean Algebra • The distinguishing factor of Boolean algebra is that it deals only with the study of binary variables. • Most commonly Boolean variables are presented with the possible values of 1 ("true") or 0 ("false"). • Variables can also have more complex interpretations, such as in set theory. Boolean algebra is also known as binary algebra. •
4. 4. Understanding Boolean Algebra • Boolean algebra is different from elementary algebra as the latter deals with numerical operations and the former deals with logical operations. • Elementary algebra is expressed using basic mathematical functions, such as addition, subtraction, multiplication, and division, whereas Boolean algebra deals with conjunction, disjunction, and negation.
5. 5. Understanding Boolean Algebra • The concept of Boolean algebra was first introduced by George Boole in his book, The Mathematical Analysis of Logic, and further expanded upon in his book, An Investigation of the Laws of Thought. • Since its concept has been detailed, Boolean algebra's primary use has been in computer programming languages. • Its mathematical purposes are used in set theory and statistics.
6. 6. Rule in Boolean Algebra
7. 7. Boolean Laws • Commutative law • Associative law • Distributive law • AND law • OR law • INVERSION law • De Morgan's Theorems
8. 8. Commutative law • Any binary operation which satisfies the following expression is referred to as commutative operation. • Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.
9. 9. Associative law • This law states that the order in which the logic operations are performed is irrelevant as their effect is the same.
10. 10. Distributive law • Distributive law states the following condition.
11. 11. AND law • These laws use the AND operation. • Therefore they are called as AND laws.
12. 12. OR law • These laws use the OR operation. • Therefore they are called as OR laws.
13. 13. INVERSION law • This law uses the NOT operation. • The inversion law states that double inversion of a variable results in the original variable itself.
14. 14. De Morgan's Theorems Theorem 1 The left hand side (LHS) of this theorem represents a NAND gate with inputs A and B, whereas the right hand side (RHS) of the theorem represents an OR gate with inverted inputs.
15. 15. De Morgan's Theorems Theorem 2  The LHS of this theorem represents a NOR gate with inputs A and B, whereas the RHS represents an AND gate with inverted inputs.
16. 16. Boolean Function • Boolean algebra deals with binary variables and logic operation. • A Boolean Function is described by an algebraic expression called Boolean expression which consists of binary variables, the constants 0 and 1, and the logic operation symbols. Consider the following example. Here the left side of the equation represents the output Y. So we can state equation no. 1
17. 17. Truth Table Formation • A truth table represents a table having all combinations of inputs and their corresponding result. • It is possible to convert the switching equation into a truth table. • For example, consider the following switching equation
18. 18. Truth Table Formation • The output will be high (1) if A = 1 or BC = 1 or both are 1. The truth table for this equation is shown by Table (a). • The number of rows in the truth table is 2n where n is the number of input variables (n=3 for the given equation). • Hence there are 23 = 8 possible input combination of inputs