1. BOOLEAN ALGEBRA

2. 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.
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3. 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”.
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4. 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.
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5. 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
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6. 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.
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7. 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.
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8. Possible Combination : YX Y Z
Now let’s draw the truth table
If the result of any logical statement is always true or , 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
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9. 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
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10. 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
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11. 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
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12. Evaluation of Boolean Expression using Truth Table
X + YZ X Y Z YZ
X+YZ 1
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13. Prove using Truth Table
(x + xy) = x
(x+y)’ = x’.y’ m
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14. 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
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15. 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
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16. 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.
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17. 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
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18. 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.
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19. 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 (ii) = 1 (iii) = 1 (iv) = 1 ( draw logic gate)
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20. 3) AND Relation (Logical Multiplication) 0. 0 = 0 0. 1 = 0 1. 0 = 0 1
3) AND Relation (Logical Multiplication) = = = = 1 4) NOT Relation 0 = 1 1 = 0 (draw logic gate)
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21. 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
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22. 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
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23. 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
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24. INVOLUTION LAW States that (A’)’ = A
is also called as double inversion law X X’
(X’)’ 1
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25. COMPLEMENTARITY LAW States that x + x’ =1 x . x’ = 0 X X’ 1 X X’ X.X’X X’
X.X’ 1
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fallacy
tautology

26. COMMUTATIVE LAW States that A B A+B B+A 1 A B AB BA 1 5/11/20191 A B AB BA 1
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27. 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
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28. DISTRIBUTIVE LAW
States that
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29. TRUTH TABLE A B C B+C AC AB A(B+C) AB+AC 1 5/11/20191
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30. 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 & Y = 1 A + AC+ BC // PROPERTIES OF 0 & 1 X.1 =1 A(1+C) + BC // PROPERTIES OF 0 & Y = 1 A+BC // PROPERTIES OF 0 & 1 X.1 =1
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31. ABSORPTION LAW
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35. Multiplying both the sides with x+y
DEMORGANS LAW
States that
Multiplying both the sides with x+y
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36. (xy)’ =x’ + y’ X Y XY (XY)’ X’ Y’ (X’+Y’) 1 5/11/20191
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