1. Boolean Algebra

2. 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.

3. Components of Truth table
TAUTOLOGY:-If result of any logical statement or expression is always TRUE or 1, it is called Tautology.
FALLACY : - If result of any logical statement or expression is always FALSE or 0, it is called Fallacy.

4. Main operator
AND (.)
OR (+)

5. AND (.) X Y
X.Y 1 1 1 1 1

6. OR (+) X Y
X . Y 1 1 1 1 1 1 1

7. BASIC LOGIC GATES
Logic Gate :- A Gate is simply an electronic circuit which operates on one or more signals to produce an output signal.
There are three types of logic gates:-
Invertor(NOT gate)
OR gate
AND gate
Invertor (NOT gate) :- An invertor(NOT gate) is a gate with only one input signal and one output signal. The output signal is always the opposite of the input state.

8. OR Gate:-The OR Gate has two or more input signals but only one output signal. If anyone input signal is high then output signal will be high.
AND Gate: - The AND Gate has two or more input signals but only one ossutput signal. If anyone input signal is low then output signal will be low. Otherwise high.

9. BASIC POSTULATES OF BOOLEAN ALGEBRA
If x != 0 then x=1 and if x!=1 then x=0
OR Relations(logical Addition)
0+0=0
0+1=1
1+0=1
1+1=1
And Relations(Logical Multiplication)
0.0=0
0.1=1
1.0=1
1.1=1
IV. Complement rules
0’ = 1
1’ = 0

10. PRINCIPLE OF DUALITY
This states that starting with a Boolean relation another Boolean relation can be derived by
Changing each OR sign to an AND sign.
Changing each AND sign to OR sign.
Replacing each 0 by 1 and each 1 by 0.
The derived relation using duality principal is called dual of original expression

11. X X X+Y X Y X.Y 1 1 1 1 1 1 BASIC THEOREMS OF BOOLEAN ALGEBRA
Properties of 0 and 1
0+x=x
1+x=1
0.x=0
1.x=x
Indempotence law
(a) x+x=x
(b) x.x=x
Proof (a) x +x=x
Proof (b) x .x=x X X
X+Y X Y
X.Y 1 1 1 1 1 1

12. Involution x’’=x Complementarity law (a) x+x’=1 (b) x.x’= 0
Cummtative law
x + y= y+x
x.y=y.x
The associative law
(i) X+(Y+Z) = (X+Y)+Z (ii) X(YZ) =(XY)Z

13. The distributive law
(i) X(Y+Z) = XY+XZ (ii) X+YZ =(X+Y)(X+Z)
The algebraic proof of law X+YZ=(X+Y)(X+Y)
(X+Y)(X+Z)=XX+XZ+XY+YZ
=X+XZ+XY+YZ
=X+XY+XZ+YZ
=X(1+Y)+Z(X+Y)
=X.1+Z(X+Y)
=X+XZ+YZ
=X+YZ
Hence proved.

14. Demorgan’s First Theorem 1.(X+Y)’=X’Y’ Proof (X+Y)+(X’Y’)=1
Absorption law
X+XY = X (ii) X(X+Y) =X
DEMORGAN’S THEOREMS
Demorgan’s First Theorem
1.(X+Y)’=X’Y’
Proof
(X+Y)+(X’Y’)=1
(X+Y)+X’Y’=((X+Y)+X’).((X+Y)+Y’)
=(X+X’+Y).(X+Y+Y’)
=(1+Y).(X+1)
=1.1 =1
(X+Y).(X’Y’)=0
(X+Y).(X’Y’)=X’Y’.(X+Y)
=X’Y’X+X’Y’Y
=0.Y+X’.0 =0
Hence proved

15. Demorgan’s Second Theorem
(X.Y)’=X’+Y’
Proof
XY+(X’+Y’)=1
=(X’+Y’)+XY
=(X’+Y’+X).(X’+Y’+Y)
=(X+X’+Y’).(X’+Y+Y’)
=(1+Y’).(X’+1)
=1.1 =1
XY. (X’+Y’)=0
=XY.( X’+Y’) ->X(Y+Z)=XY+XZ
=XYX’+XYY’
=0.Y+X.0
=0+0 =0

16. Derivation of Boolean expression
1) Minterms: - Minterms is a product of all the literals (with or without the bar) within the logic system.
Example: Convert X+Y to minterm
X+Y = X.1+Y.1
= X.(Y+Y’)+Y.(X+X’)
= XY+XY’+XY+X’Y
= XY+XY’+X’Y
2) Maxterms: - Maxterm is a sum of all the literals (with or without the bar) within the logic system.
Example: If the value of the variables are X=0, Y=1 and Z=1 then the Maxterm will be X+Y’+Z’

17. 3) Canonical Expression: - Boolean expreesion composed entirely either of minterms and Maxterm is referred to as canonical expression. Canonical expression can be represented in two forms:-
Sum – of – products (SOP) form
ii) Product – of – Sum (POS) form
Canonical sum of product form:- When a Boolean expression is represented purely as sum of minterms or product terms, it is said to be canonical product form

18. Canonical Product of sum form: - When a Boolean expression is represented purely as product of maxterms, it is said to be in canonical product of sum form of expression.
Minimization of Boolean expression
Algebraic method
Example : - Reduce X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’
= X’(Y’Z’ + YZ’) + X(Y’Z’+YZ’)
=X’(Z’(Y’+Y)) + X(Z’(Y’+Y))
=X’(Z’.1)+X(Z’.1)
=X’Z’+XZ’
=Z’(X’+X’)
=Z’.1
=Z’

19. Simplification using karnaugh maps
Karnaugh map: - Karnaugh map or K – map is a graphical display of the fundamental products in a truth table.
K – map is nothing but a rectangle made up of certain number of squares represents a Maxterm or minterm
SOP reduction using K-maps : -
For a function of n variables, there would be a map of 2n squares each represents a minterm.

20. POS reduction using K-maps : -
For a function of n variables, there would be a map of 2n squares each represents a maxterm.

21. More about Logic Gates
NOR Gates:-
The NOR gates has two or more input signals but only one input signal. If all inputs are 0 (i.e., low), then the output signal is 1 (high).
NAND Gate:-
The NAND gate has two or more input signals but only one output signal. If all of the inputs are (high), then the output produced is 0 (low).
the input combination has even number of 1’s.

22. XOR Gate (Exclusive OR gate):-
The XOR gate can also have two or more inputs but produces one output signal. Exclusive-OR gate different from OR gate. OR gate produces output 1 for any input combination having one or more 1’s, but XOR gate produces output 1 for only those input combinations that have odd number 1’s.
XNOR Gate (Exclusive NOR gate)
The XNOR Gate is logically equivalent to an inverted XOR i.e., XOR gate followed by a NOT gate (inventor). Thus XNOR produces 1 (high) output when