ERROR Compile time error, Runtime error

error

ERROR

            “Error is the mistake that makes a program not function properly”. In programming there are two types of error which can raise during programming that’s are

  ð  Compile time error

  ð  Runtime error

Read More

HALF ADDER AND FULL ADDER

HALF ADDER

            Half adder is like a small circuit that performs an addition tasks on only two inputs 0 and 1 for each input and gives us two outputs one is sum and the second is carryout number. Which consist of and exclusive OR gate that and AND gate including two inputs, where X-OR gate perform Sum and AND gate gives carry out number, which is more explain by logic diagram, logic symbol and truth table.

LOGIC DIAGRAM

half adder logic diagram

            

           Where the A and B is input and Sum = AÅB output of XOR and the Carry out number is A x B output of AND gate.

LOGIC SYMBOL:

halfadder logic symbol

TRUTH TABLE

A	B	AÅB	A x B
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

 FULL ADDER

            Process is same like half adder that perform addition and full adder also perform addition where the main difference between them is the number of input, where half adder have only two input and full adder have three inputs A, B and Cin. It also has two outputs one is sum and second is carryout number. See the logic diagram, logic symbol and truth table for understand to deep.

LOGIC SYMBOL 

fulladder logic diagram

          In this series three inputs are A, B and Cin where the final results are Sum and Carryout number as shown in figure. In full adder we used 5 gates two AND gate two XOR gate and one OR gate.

LOGIC SYMBOL

            Arrangement of two half adder to form a full adder in such a way that it look like as show

full adder logic symbol

TRUTH TABLE

Inputs	Outputs
Cin A B 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1	Sum Carryout 0 0 1 0 1 0 0 1 1 0 0 1 0 1 1 1

Read More

K-Map (for Boolean expression simplification)

K-MAP:

   " K map provide a systematical method for the simplification of Boolean expressions. "

Basically there are two methods by which we can simplify the Boolean expression first one is from 12 laws and rules of Boolean algebra and the second is by using K-map. By using K-map we can easily simplify the Boolean than laws and rule of Boolean algebra.

  We have three type of K-map according to the width of the variable. That’s.

1.    3 variable K-map

2.    4 variable K-Map

3.    5 variable K-map

Read More

12 LAWS AND RULE OF BOOLEAN ALGEBRA

basic rules of boolean algebra

BOOLEAN ALGEBRA'S RULES AND LAWS:

  A English mathematician, philosopher and logician George Boole 1854 given a mathematical theories and few logical algebra named as Boolean Algebra..which consist of 12 laws or rule which briefly described Boolean Algebra which are:

            

1.    A + 0 = A

2.    A + 1 = 1

3.    A . 0  = 0

4.    A . 1  = A

5.    A + A = A

6.    A + Ā = 1

7.    A . A = A

8.    A . Ā= 0

9.    ﬢ(ﬢA) = A

10. A + A B = A

11. A + ĀB = A + B

12. (A + B)(A + C) = A + BC

Derivation of Laws Using Gates And Truth Table:

RULE # 1 (  A + 0 = A) 

A + 0 = A

A	0	A+0
0	0	0
1	0	1

RULE # 2 (A + 1 = 1)

A + 1 = 1

A	1	A+1
0	1	1
1	1	1

RULE # 3 (  A . 0  = 0)

A .0 =0

A	0	A . 0
0	0	0
1	0	0

RULE# 4 (A . 1  = A)

A. 1= A

A	1	A . 0
0	1	A
1	1	A

RULE #5 ( A + A = A)

A + A = A.

A	A	A +A
0	0	A
1	1	A

RULE # 6(A + Ā = 1)

A + Ā = 1

A	Ā	A+ Ā=1
0	1	1
1	0	1

RULE # 7(A . A = A)

A.A=A

A	A	A .A
0	0	A
1	1	A

RULE # 8 (  A . Ā= 0)

A . Ā= 0

A	Ā	A . Ā
0	1	0
1	0	0

RULE # 9 (ﬢ(ﬢA) = A)

A Double not = A

RULE #10 ( A + A B = A)

= A+ AB

= A (1 + B)

= A (1)             (from rule no 2 A + 1= 1, replace the variable A by B so  B+1 or 1+B=1)

= A

A + AB = A

RULE # 11( A + ĀB = A + B)

= A + ĀB

= A + AB + ĀB   (from rule no 10,A+AB=A)

= A + B(A +Ā )

= A + B(1)          (from rule no 6, A +Ā = 1)

= A + B              (which is equal to right hand side)

 A + ĀB = A + B

RULE#12(. (A + B)(A + C) = A + BC)

= (A + B)(A + C)

= AA + AC + AB + BC

= A + AC + AB + BC        (from rule no 7 A.A = A)

= A(1 + C) + AB + BC

= A (1) + AB +BC             (from rule no 2, 1+C=1)

= A + AB + BC

= A + BC                           (from rule no 10 , A + AB = A) hence it is equal to right hand side

(A+B)(A+C)=A+AB

                                           physics label                                                                                                  

Read More