BOOLEAN ALGEBRA

INTRODUCTION Aristotle - constructed a complete System of Formal logic and wrote six famous works on the subject. It contributed to the Organisation of Man’s Reasoning. George Boole – Manipulated this and arrived at his own mathematical system of logic. His revolutionary paper – “ An investigatioon of the laws of the thought” published in 1854 led to the development of Boolean Algebra. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

In 1938 Claude E Shannon wrote a paper titled “ A Symbolic Analysis of Relay Switching Circuits” which applied Boolean Algebra to solve relay logic problems, which dealt with binary values. Boolean Algebra effectively deals with binary values. Thus it came to be known as “Switching Circuits”. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Every Day we take decisions The decision which results in either Yes or No is called Binary Decision called Truth Values. The Vairables that store these Truth Values are called Logical values or Binary Valued Variables. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Boolean Algebra is suitable for Digital Circuits The Principle behind Boolean Algebra is that answers to its equations are either True or False. The Statement which may either be True or False is called Proposition. Boolean Algebra is also known as Switching Algebra Propostion Algebra Two-State Algebra Logical values T or F are called Logical Constants 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Logical Functions or Compound Statements : Algebraic variables a,b,c or x,y,z are combined with mathematical operators to form algebraic expressions eg., a+b+c Likewise, logical statements are combined with logical operators like AND, OR, NOT to form compound statement or logical functions. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

What is Truth Table? Truth table is a table which represents all the possible values of logical variables/statements along with all the possible results of the given combinations of values. Statement 1 : I want Tea Statement 2 : Tea is readily available Result : I ‘ll have tea. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Possible Combination : Y X Y Z Now let’s draw the truth table If the result of any logical statement is always true or 1, it is called Tautology, and if the result is always false or 0 then it is called Fallacy Statement 1 T F Statement 1I RESULT X Y Z 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Logical Operator NOT OPERATOR Operates on a single variable and the operation performed by NOT operator is called Complementation. x’ means complement of x Truth Table X X’ 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

OR OPERATOR Operation is called Logical addition and the symbol used is +. Therefore, + does not have normal meaning but logical addition or logical OR symbol. X + Y denotes X OR Y X Y X+Y 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

AND OPERATOR Operation is called Logical multiplication and the symbol used is . Therefore, . does not have normal meaning but logical multiplication or logical AND symbol. X . Y denotes X AND Y X Y X.Y 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Evaluation of Boolean Expression using Truth Table X + YZ X Y Z YZ X+YZ 1 S KIRAN PGT(COMP) KV CLRI

Prove using Truth Table (x + xy) = x (x+y)’ = x’.y’ m 5/11/2019 S KIRAN PGT(COMP) KV CLRI

BASIC LOGIC GATES After Shannon applied Boolean Algebra for Tele-switching circuits, engineers realised that Boolean Algebra can be applied for Computer electronics In Computers these Boolean operations are performed by Logic Gates. What are Logic Gates ? A Gate is simply an electronic circuit which operates on one or more signals to produce an output signal 5/11/2019 S KIRAN PGT(COMP) KV CLRI

Gates are often called Logic Circuits There are 3 types of Logic Gates Gates are digital (two-state) b’cos input and output signals are either low(0) or high(1). Gates are often called Logic Circuits There are 3 types of Logic Gates Inverter (NOT) Gate OR Gate AND Gate 5/11/2019 S KIRAN PGT(COMP) KV CLRI

NOT GATE An inverter also called NOT gate, is a gate with only one input signal and output signal; the output state is always the opposite of the input state. The output is also called complement of the input. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

OR GATE OR Gate has two or more input signal but only one output signal. If any of the input signal is 1(high) the output signal is 1(high), if all inputs are 0 then the result is 0 5/11/2019 S KIRAN PGT(COMP) KV CLRI

AND GATE AND Gaete can have two or more input signals and produce an output signal. When all inputs are high ie., 1 then the output is 1 otherwise 0. 5/11/2019 S KIRAN PGT(COMP) KV CLRI

BASIC POSTULATES IN B.A 1) If x ≠ 0 then x = 1 and if x ≠ 1 then x = 0 2) OR Relations (logical Addition) (i) 0 + 0 =0 (ii) 0 + 1 = 1 (iii) 1 + 0 = 1 (iv) 1 + 1 = 1 ( draw logic gate) 5/11/2019 S KIRAN PGT(COMP) KV CLRI

3) AND Relation (Logical Multiplication) 0. 0 = 0 0. 1 = 0 1. 0 = 0 1 3) AND Relation (Logical Multiplication) 0 . 0 = 0 0 . 1 = 0 1 . 0 = 0 1 . 1 = 1 4) NOT Relation 0 = 1 1 = 0 (draw logic gate) 5/11/2019 S KIRAN PGT(COMP) KV CLRI

PRINCIPLES OF DUALITY States that starting with a Boolean expression , another Boolean expression can be derived by Changing each OR sign (+) to AND (.) sign Changing each AND sign (.) to OR(+) sign Replacing each 0 by 1 and each 1 by 0 For eg., from Postulate 2 postulate 3 can be derived 5/11/2019 S KIRAN PGT(COMP) KV CLRI

BASIC THEOREMS OF B.A Properties of 0 and 1 0 + X = X 1 + X = 1 (LOGIC GATE) X R 1 1 X R X R 1 1 X R 5/11/2019 S KIRAN PGT(COMP) KV CLRI

IDEMPOTENCE LAW States that (i) x + x =x (ii) x . x = x (i) and (ii) are duals of each other X X + X 1 X X .X 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

INVOLUTION LAW States that (A’)’ = A is also called as double inversion law X X’ (X’)’ 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

COMPLEMENTARITY LAW States that x + x’ =1 x . x’ = 0 X X’ 1 X X’ X.X’ X X’ X.X’ 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI fallacy tautology

COMMUTATIVE LAW States that A B A+B B+A 1 A B AB BA 1 5/11/2019 1 A B AB BA 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

ASSOCIATIVE LAW Since (ii) is the dual of (i), hence it is also proved States that Since (ii) is the dual of (i), hence it is also proved A B C B+C A+B A+(B+C) (A+B)+C 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

DISTRIBUTIVE LAW States that 5/11/2019 S KIRAN PGT(COMP) KV CLRI

TRUTH TABLE A B C B+C AC AB A(B+C) AB+AC 1 5/11/2019 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

ALGEBRAIC PROOF A + BC = (A+B)(A+C) Taking R.H.S AA +AC+ AB +BC A + AB + AC +BC // IDEMPOTENCE LAW X.X =X A(1+ B) +AC +BC // PROPERTIES OF 0 & 1 1 + Y = 1 A + AC+ BC // PROPERTIES OF 0 & 1 X.1 =1 A(1+C) + BC // PROPERTIES OF 0 & 1 1 + Y = 1 A+BC // PROPERTIES OF 0 & 1 X.1 =1 5/11/2019 S KIRAN PGT(COMP) KV CLRI

ABSORPTION LAW 5/11/2019 S KIRAN PGT(COMP) KV CLRI

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Multiplying both the sides with x+y DEMORGANS LAW States that Multiplying both the sides with x+y 5/11/2019 S KIRAN PGT(COMP) KV CLRI

(xy)’ =x’ + y’ X Y XY (XY)’ X’ Y’ (X’+Y’) 1 5/11/2019 1 5/11/2019 S KIRAN PGT(COMP) KV CLRI