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Unit-IV: Combinational Logic Circuits ❖ Outlines of Unit - IV 1) Introduction and Laws of Boolean Algebra :- I

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Unit-IV: Combinational Logic Circuits ❖ Outlines of Unit - IV 1) Introduction and Laws of Boolean Algebra :- I Faculty: Science Course: BSc Sem: II Unit: IV Paper: II Topic: Combinational Logic Circuits Teacher: Saikumar Chirra Physics Department. Unit-IV: Combinational Logic Circuits ❖ Outlines of Unit - IV 1) Introduction and laws of Boolean Algebra :- i. Commutative ii. Associative iii. Distribution and iv. Absorptive Laws 2) De-Morgan’s laws:- 3) Dual & Complement of Boolean Function: 4) SOP & POS (or) Minterms & Maxterms: 5) Karnaugh’s (K) - Mapping: 6) Simplification of Boolean Expression:- By i. Boolean laws ii. K - maps 7) Combinational Logic Circuits: i. Half adder, ii. Full adder, iii. Half Subtractor and iv. Full Subtractor. 1. Introduction and laws of Boolean Algebra Introduction: ➢ Boolean Algebra is the mathematics we use to analyze digital gates and circuits and these gates are used as the building blocks in the design of more complex digital logic circuits. ➢ We can use these “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. ➢ Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules or laws which are used to define and reduce Boolean expressions. It is named after its inventor George Boole. ➢ The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labelled individually to represent inputs to the expression 1. Introduction and laws Continue…… ➢ The algebra that deals with binary literals and logic functions ➢ literals: Denote by letters of the alphabet. e.g. A, B, X, Y, Z (or) a, b, x, y, z ….etc ➢ Basic Logic operations on those literals: AND, OR, NOT ➢ A Boolean Expression (e.g. X+YZ) is Formed by: - Binary literals -Logic operations (operators) on the literals and constants -Constants 0,1 ➢ A Boolean Function can be described by a Boolean Equation of the form ➢ Each equation can be represented as a logic diagram ➢ A Boolean Function can be uniquely expressed as a truth table showing the mapping between each possible combination of the input literals and the output literal (n input literals = 2ncombinations) ➢ Simplest functions require the smallest number of the smallest gates. 1. Introduction and laws Continue…… George Boole, 1815 - 1864 • Born to working class parents • Taught himself mathematics and joined the faculty of Queen’s College in Ireland. • Wrote An Investigation of the Laws of Thought (1854) • Introduced binary variables • Introduced the three fundamental logic operations: AND, OR, and NOT. Copyright © 2007 Elsevier Simplifying a Logic Circuit F = A(A’ + B) The expression F simplifies to AB. Which circuit is cheaper? In order to use of Boolean Algebra we need to know some basic Laws and Principles. 1. Introduction and laws Continue…… i. Commutative Law • The order of application of two separate terms is not important • i.e. A * B = B * A ➢ The order in which two variables are AND’ed makes no difference. A . B = B . A ➢ The order in which two variables are OR’ed makes no difference. A + B = B + A ii. Associative law • This law allows the removal of brackets from an expression and regrouping of the variables. • i. e. (A * B) * C = A * (B * C) = A * B * C ➢ A + (B + C) = (A + B) + C = A + B + C (OR Associate Law) ➢ A(B.C) = (A.B)C = A . B . C (AND Associate Law) 1. Introduction and laws Continue…… iii. Distributive Law • This law permits the multiplying or factoring out of an expression. • i.e. A * B . C = A * B . A * C ➢ A.(B + C) = A.B + A.C (OR Distributive Law) ➢ A + (B.C) = (A + B).(A + C) (AND Distributive Law) iv. Absorptive Law • This law enables a reduction in a complicated expression to a simpler one by absorbing like terms. • B is irrelevant (redundant, absorbed) in this expression. • i.e. A * (A * B) = A ➢ A + (A.B) = A (OR Absorption Law) ➢ A(A + B) = A (AND Absorption Law) 1. Introduction and laws Continue…… Basic Principles of Boolean algebra • A Boolean algebra requires: – A set of values B, which contains at least two elements 0 and 1 – Two 2-argument operations + and • – A one-argument operation ' • The values and operations must satisfy the axioms below. 1. x + 0 = x 2. x • 1 = x 3. x + 1 = 1 4. x • 0 = 0 5. x + x = x 6. x • x = x 7. x + x’ = 1 8. x • x’ = 0 9. (x’)’ = x 10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y)’ = x’y’ 17. (xy)’ = x’ + y’ DeMorgan’s 2. De-Morgan’s laws:- • Augustus De Morgan was a contemporary of George Boole. He was the first professor of mathematics at the University of London. • De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. • De Morgan did not actually create the law given his name, but is credited with stating it. • De Morgan’s axioms/Laws are especially important for circuit design. • Easily generalized to n variables. 2. De-Morgan’s laws Continue…. • De Morgan's laws are a pair of transformation rules that are both valid rules of inference. • Can be proven using a Truth table or Boolean Algebra ➢ 1st Law : - When the NOT operator is applied to the AND of two variables, it is equal to the NOT applied to each of the variables with an OR in between: 퐘 = 푨 . 푩 = 푨ഥ + 푩ഥ Or (x . y)’ = x’ + y’ ✓ Pushing a bubble from the output back to the inputs puts bubbles on all gate inputs. A A Y Y B B 2. De-Morgan’s laws Continue…. ➢ 2nd Law: - When the NOT operator is applied to the OR of two variables, it is equal to the NOT applied to each of the two variables with an AND in between: 퐘 = 푨 + 푩 = 푨ഥ . 푩ഥ Or (x + y)’ = x’ . y’ ✓ Pushing bubbles on all gate inputs forward toward the output puts a bubble on the output and changes the gate body. A A Y Y B B 3. Dual & Complement of Boolean Function: Duality ➢ The dual of any equation is always true. ➢ For any expression, the dual is formed by replacing • AND with OR, • OR with AND, • 0 with 1, and • 1 with 0. • Variables and complements are left unchanged. Ex: (AB' + C)' = (A + B').C (OR) ✓ The dual of an expression may be found by complementing the entire expression and then complementing each of the individual variables. Ex: (AB' + C)' = (AB')'.C' = (A' + B).C' Therefore, (AB' + C)D = (A + B').C 3. Dual & Complement Continue…. Duality • Look carefully at the left and right columns of axioms. • They are duals, where we’ve: – exchanged all AND operations with OR operations, and – exchanged all 0s with 1s. • The dual of any equation is always true. 1. x + 0 = x 2. x • 1 = x 3. x + 1 = 1 4. x • 0 = 0 5. x + x = x 6. x • x = x 7. x + x’ = 1 8. x • x’ = 0 9. (x’)’ = x 10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y)’ = x’y’ 17. (xy)’ = x’ + y’ DeMorgan’s 3. Dual & Complement Continue…. Complement of a function • When you complement an expression algebraically, you can use De-Morgan’s axiom to keep “pushing” the complement operators inwards: If f(x,y,z) = x(y’z’ + yz) then find f’ ?? Sol: f’(x,y,z) = ( x(y’z’ + yz) )’ [ complement both sides ] = x’ + (y’z’ + yz)’ [ because (xy)’ = x’ + y’ ] = x’ + (y’z’)’ (yz)’ [ because (x + y)’ = x’ y’ ] = x’ + (y + z)(y’ + z’) [ because (xy)’ = x’ + y’] 3. Dual & Complement Continue…. Complement of a function • The magenta axioms deal with the complement operation, which doesn’t exist in normal algebra. • Most of them almost make sense if you translate them into English: – “It is raining or it is not raining” is always true (x + x’ = 1) – “It is raining and it is not raining” can never be true (x • x’ = 0) – “I am not not handsome” means “I am handsome” ((x’)’ = x) 1. x + 0 = x 2. x • 1 = x 3. x + 1 = 1 4. x • 0 = 0 5. x + x = x 6. x • x = x 7. x + x’ = 1 8. x • x’ = 0 9. (x’)’ = x 10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y)’ = x’y’ 17. (xy)’ = x’ + y’ DeMorgan’s 3. Dual & Complement Continue…. Complement of a function • The complement of a function always outputs 0 where the original function output 1, and 1 where the original produced 0. • In a truth table, we can just exchange 0s and 1s in the output column. – On the left is a truth table for f(x,y,z) = x(y’z’ + yz) – On the right is the table for the complement, denoted f’(x,y,z). x y z f(x,y,z) x y z f’(x,y,z) 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 3.
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