Boolean connectives

Propositional Logic and Hardware

Connectives used in propositional logic

Negation (NOT) : Represented by ~ or ‘ or a -(bar) sign. It is an operator that affacts a single statements only and does not join two or more statements. it  it is also called as unary connective. It’s output is always inverse it input.

Conjunction (AND) : Represented by the symbol . or ^. It works on more than one proposition and the output is true when all the propositions are true. Conjunction means both the proposition are true.

Disjunction (AND) : Represented by the symbol + or v. It works on more than one proposition and the output is true when any of the proposition is true. Disjunction means either of the proposition is true.

Conditional (if ... then or Implication) : Represented by the symbol  or  or . Implication means if one proposition is true then other proposition is true.

Bi-conditional (if and only if or Equivalence) : Represented by the symbol or  . Equivalence means both propositions are true or both are false.

Truth Table and Truth Values : A Truth Table is a complete list of possible truth values of a proposition and the Truth Values are defined as truth or falsity of a proposition.

Truth Tables of all the connectives : 

Related Terms

Contingencies : The proposition that have some combination of 1’s and 0’s in their truth table column, are called Contingencies.

Tautology : The proposition that have all 1’s in their truth table column, are called Tautology.

Contradictions : The proposition that have all 0’s in their truth table column, are calledContradictions.

Consistent Statement : Two statements are consistent if and only if their conjunction is not a contradiction.

Converse : The converse of a conditional proposition is determined by interchanging the antecedent (The first proposition) and consequent (The second proposition) of given conditional.  if  X => Y then its converse is Y => X.

Inverse : The inverse of a conditional proposition is another conditional having negated antecedent and consequent.  if  X => Y then its inverse is X' => Y'.

Contrapositive : The contrapositive of a conditional is formed by creating another conditional that takes its antecedent as negated consequent of earlier condition and consequent as negated antecedent of earlier conditional. If  X => Y then its contrapositive is Y' => X'

Gaurav Nigam

Read More

Boolean simplification

Simplification of Boolean Expressions

Here are some examples of Boolean algebra simplifications. Each line gives a form of the expression, and the rule or rules used to derive it from the previous one. Generally, there are several ways to reach the result. Here are some of the examples for you as guideline :

• Simplify: AB(A + B)(B + B):

Expression	Rule(s) Used
AB(A + B)(B + B)	Original Expression
AB(A + B)	Complement law, Identity law.
(A + B)(A + B)	DeMorgan's Law
A + BB	Distributive law. This step uses the fact that or distributes over and. It can look a bit strange since addition does not distribute over multiplication.
A	Complement, Identity.

• Simplify: (A + C)(AD + AD) + AC + C:

Expression	Rule(s) Used
(A + C)(AD + AD) + AC + C	Original Expression
(A + C)A(D + D) + AC + C	Distributive.
(A + C)A + AC + C	Complement, Identity.
A((A + C) + C) + C	Commutative, Distributive.
A(A + C) + C	Associative, Idempotent.
AA + AC + C	Distributive.
A + (A + T)C	Idempotent, Identity, Distributive.
A + C	Identity, twice.

• Simplify: A(A + B) + (B + AA)(A + B):

Expression	Rule(s) Used
A(A + B) + (B + AA)(A + B)	Original Expression
AA + AB + (B + A)A + (B + A)B	Idempotent (AA to A), then Distributive, used twice.
AB + (B + A)A + (B + A)B	Complement, then Identity. (Strictly speaking, we also used the Commutative Law for each of these applications.)
AB + BA + AA + BB + AB	Distributive, two places.
AB + BA + A + AB	Idempotent (for the A's), then Complement and Identity to remove BB.
AB + AB + AT + AB	Commutative, Identity; setting up for the next step.
AB + A(B + T + B)	Distributive.
AB + A	Identity, twice (depending how you count it).
A + AB	Commutative.
(A + A)(A + B)	Distributive.
A + B	Complement, Identity.

Read More

Simplification Of Boolean Functions Using K-Map

Follow the Link to check out for the solution

Read More

Syllabus For Class XII Computer Science

Class XII (Theory) – C++

Duration: 3 hours Total Marks: 70

Unit 1: Object Oriented Programming in C++ (50 Theory + 40 Practical) Periods

REVIEW: C++ covered In Class – XI,

Object Oriented Programming: Concept of Object Oriented Programming – Data hiding, Data

encapsulation, Class and Object, Abstract class and Concrete class, Polymorphism

(Implementation of polymorphism using Function overloading as an example in C++); Inheritance,

Advantages of Object Oriented Programming over earlier programming methodologies,

Implementation of Object Oriented Programming concepts in C++: Definition of a class,

Member of a class – Data Members and Member Functions (methods), Using Private and Public

visibility modes, default visibility mode (private); Member function definition: inside class

definition and outside class definition using scope resolution operator (::); Declaration of objects

as instances of a class; accessing members from object (s), Objects as function arguments–pass by

value and pass by reference;

Constructor and Destructor: Constructor: special characteristics, declaration and definition of a

constructor, default constructor, overloaded constructors, copy constructor, constructor with

default arguments;

Destructor: Special Characteristics, declaration and definition of destructor;

Inheritance (Extending Classes): Concept of Inheritances, Base Class, Derived classes, protected

visibility mode; Single level inheritance, Multilevel inheritance and Multiple inheritance, Privately derived, publicly derived and Protectedly derived class, accessibility of members from objects and

within derived class (es);

Data File Handling: Need for a data file, Types of data files – Text file and Binary file;

Text File: Basic file operations on text file: Creating/Writing text into file, Reading and

Manipulation of text from an already existing text File (accessing sequentially).

Binary File: Creation of file, Writing data into file, Searching for required data from file,

Appending data to a file, Insertion of data in sorted file, Deletion of data from file, Modification

of data in a file;

Implementation of above mentioned data file handling in C++;

Components of C++ to be used with file handling:

Header file: fstream.h; ifstream, ofstream, classes;

Opening a text file in in, out, and app modes;

Using cascading operators (>><<) for writing text to the file and reading text from the file; open

(), get (), read () put (), write(), getline() and close() functions; Detecting end-of-file (with or

without using eof() function), tellg(), tellp(), seekg().seekp();

Pointers:

Introduction to Pointer, Declaration and Initialization of Pointer; Dynamic memory allocation/deallocation

operators: new, delete; Pointers and Arrays: Array of Pointers, Pointer to an array (1

dimensional array), Function returning a pointer, Reference variables and use of alias; Function

call by reference. Pointer to structure: De-reference/Deference operator: *, ->; self referencial

structure;

Unit 2: Data Structures (42 Theory + 36 Practical) Periods

Introduction to data structure- array, stack queues primitive and non-primitive data structure,

linear and non-linear structure, static and dynamic data structure.

Arrays:

One and two Dimensional arrays: Sequential allocation and address calculation;

One dimensional array: Traversal, Searching (Linear, Binary Search), Insertion of an element in an

array, deletion of an element from an array, Sorting (Insertion, Selection, Bubble)

Two-dimensional arrays: Traversal Finding sum/difference of two NxM arrays containing numeric

values, Interchanging Row and Column elements in a two dimensional array;

Stack (Array and Linked implementation of Stack):

Introduction to stack (LIFO_Last in First out Operations)

Operations on stack (PUSH and POP) and its Implementation in C++, Converting expressions from

INFIX to POSTFIX notation and evaluation of Postfix expression;

Queue: (Array and Linked Implementation)

Introduction to Queue (FIFO – First in First out operations)

Operations on Queue (Insert and Delete and its Implementation in C++, circular queue using array.

Read More

Sorting Techniques

Sorting Techniques

/* Sorting ==> Bubble Sort technique */

#include<iostream.h>

#include<conio.h>

#define max 100

void main()

{

int arr[max], len, i, j, temp;

cout<<"Entre the size of array :\t";

cin>>len;

for(i=0; i<len; i++)

{

cout<<"Enter element "<<i+1<<" : ";

cin>>arr[i];

}

cout<<"\n Original array is :\n";

for(i=0; i<len; i++)

{

cout<<arr[i]<<" , ";

}

//Sorting starts here

for(i=0; i<len; i++)

{

for(j=0; j<len-1; j++)

{

if(arr[j] >arr[j+1])

{

temp = arr[j];

arr[j] = arr[j+1];

arr[j+1] = temp;

}

}

}

       //Sorting over

//Display sorted list

cout<<"Sorted array is :\n";

for(i=0; i<len; i++)

{

cout<<arr[i]<<"  ";

}

getch();

}

/* Sorting ==> Selection Sort technique */

#include<iostream.h>

#include<conio.h>

#define max 100

void main()

{

int arr[max], len, i, j, temp;

cout<<"Entre the size of array :\t";

cin>>len;

for(i=0; i<len; i++)

{

cout<<"Enter element "<<i+1<<" : ";

cin>>arr[i];

}

cout<<"\n Original array is :\n";

for(i=0; i<len; i++)

{

cout<<arr[i]<<" , ";

}

//Sorting starts here

         for(i=0; i<len-1; i++)

 {

               int low = arr[i];

               int p = a[i];

for(j=i+1; j<len; j++)

{

if(low>arr[i])

{

low = arr[i];

                                p = i;

}

}

         if(p != i)

         {

                temp = arr[i];

                arr[i] = arr[p];

                arr[p]=temp;

}

       }

       //Sorting over

//Display sorted list

cout<<"Sorted array is :\n";

for(i=0; i<len; i++)

{

cout<<arr[i]<<"  ";

}

getch();

}

Read More

Project programs for cbse 2016

Project Programs for CBSE Class XII

Q1. WAP to generate an Anti-Spiral matrix.

Solution :

#include<iostream.h>
#include<conio.h>
void main()
{
clrscr();
int a[20][20],x,y,n,s,p;
cout<<"Enter size (even no.):";
cin>>s;
cout<<"Enter number:";//entry of starting no
cin>>n;
s--;
p=s/2;
for(x=p;x>=0;x--)//filling of elements in required order
{
  for(y=x;y<=(s-x);y++)
    a[x][y]=n++;
  for(y=(x+1);y<=(s-x);y++)
    a[y][s-x]=n++;
  for(y=s-(x+1);y>=x;y--)
    a[s-x][y]=n++;
  for(y=s-(x+1);y>x;y--)
    a[y][x]=n++;
}
cout<<endl;
for(x=0;x<=s;x++)//display of matrix after filling
{
  for(y=0;y<=s;y++)
    cout<<a[x][y]<<"\t";
  cout<<endl<<endl;
}
getch();
}


/*     OUTPUT

Enter size (even no.):4
Enter number:5

9       10      11      12

20      5       6       13

19      8       7       14

18      17      16      15

*/

Q2. WAP to find future date.

Solution :

#include<iostream.h>
#include<conio.h>
void main()
{
clrscr();
int d,m,y,da;
cout<<"enter date in dd-mm-yyyy format:";
cin>>d;cout<<"-";cin>>m;cout<<"-";cin>>y;
cout<<"enter no. of days:";
cin>>da;
while(da>0)
{
if(m==1 || m==3 || m==5 || m==7 || m==8 || m==10 || m==12)
{
if(d<31)
{
 d++;
 da--;
 continue;
}
if(d==31 && m!=12)
{
 m++;
 d=1;
 da--;
 continue;
}
if(d==31 && m==12)
{
 y++;
 m=1;
 d=1;
 da--;
 continue;
}
}
if(m==4 || m==6 || m==9 || m==11)
{
if(d<30)
{
 d++;
 da--;
 continue;
}
if(d==30)
{
 m++;
 d=1;
 da--;
 continue;
}
}
if(m==2)
{
  if(y%4==0)
  {
if(d<29)
{
 d++;
 da--;
 continue;
}
if(d==29)
{
 m++;
 d=1;
 da--;
 continue;
}
  }
  else
  {
if(d<28)
{
 d++;
 da--;
 continue;
}
if(d==28)
{
 m++;
 d=1;
 da--;
 continue;
}
  }
}
}
cout<<"the future date= "<<d<<"-"<<m<<"-"<<y;
getch();
}


/*     OUTPUT

enter date in dd-mm-yyyy format:11 05 2000
--enter no. of days:200
the future date= 27-11-2000


*/

Q3.WAP to find the number of days between two dates.

Solution :

#include<iostream.h>
#include<conio.h>
void main()
{
clrscr();
int d,m,y,da=0,d2,m2,y2;
cout<<"enter 1st date in dd-mm-yyyy format:";
cin>>d;cout<<"-";cin>>m;cout<<"-";cin>>y;
cout<<"enter 2nd date in dd-mm-yyyy format:";
cin>>d2;cout<<"-";cin>>m2;cout<<"-";cin>>y2;
while(d!=d2 || m!=m2 || y!=y2)
{
da++;
if(m==1 || m==3 || m==5 || m==7 || m==8 || m==10 || m==12)
{
if(d<31)
{
 d++;
 continue;
}
if(d==31 && m!=12)
{
 m++;
 d=1;
 continue;
}
if(d==31 && m==12)
{
 y++;
 m=1;
 d=1;
 continue;
}
}
if(m==4 || m==6 || m==9 || m==11)
{
if(d<30)
{
 d++;
 continue;
}
if(d==30)
{
 m++;
 d=1;
 continue;
}
}
if(m==2)
{
  if(y%4==0)
  {
if(d<29)
{
 d++;
 continue;
}
if(d==29)
{
 m++;
 d=1;
 continue;
}
  }
  else
  {
if(d<28)
{
 d++;
 continue;
}
if(d==28)
{
 m++;
 d=1;
 continue;
}
  }
}
}
cout<<"The number of days="<<da;
getch();

}

Q4. WAP to find the number of vowel pairs in an entered string.

Solution :

#include<iostream.h>
#include<conio.h>
#include<string.h>
void main()
{
clrscr();
char s[100],v[10]="AEIOUaeiou";//string of all vowels to be checked
int l,a[5][5],x,y,c1,c2,f,p1,p2;
for(x=0;x<5;x++)
{
  for(y=0;y<5;y++)
    a[x][y]=0;
}
cout<<"Enter sentence:";
cin.getline(s,100);
l=strlen(s);
for(x=0;x<l-1;x++)
{
  f=0;
  c1=s[x];
  c2=s[x+1];
  for(y=0;y<10;y++)
  {
    if(c1==v[y])
    {
      p1=y;
      for(y=0;y<10;y++)
      {
if(c2==v[y])
{
 p2=y;
 f=1;
}
      }
    break;
    }
  }
  if(f==1)
  {
    if(p1>4)
      p1-=5;
    if(p2>4)
      p2-=5;
    a[p1][p2]++;
  }
}
cout<<"\n\nVovel pairs are:\n\n\n";
cout<<"\t";
for(x=0;x<5;x++)
  cout<<v[x]<<"\t";
cout<<endl<<endl;
for(x=0;x<5;x++)
{
  cout<<v[x]<<"\t";
  for(y=0;y<5;y++)
    cout<<a[x][y]<<"\t";
  cout<<endl<<endl;
}
getch();

}

Q5. WAP to enter a number and print it in words.

Solution :

#include<iostream.h>
#include<conio.h>
void main()
{
clrscr();
int n,r;
char na1[9][5]={"one","two","three","four","five","six","seven","eight","nine"};
char na2[8][7]={"twenty","thirty","fourty","fifty","sixty","seventy","eighty","ninty"};
char na3[10][9]={"ten","eleven","twelve","thirteen","fourteen","fifteen","sisteen","seventeen","eighteen","nineteen"};
cout<<"Enter number:";
cin>>n;
if(n>=1000)//checking thousand'th digit for n>1000
{
r=n/1000;
n=n%1000;
r--;
cout<<na1[r]<<" thousand ";
}
if(n>=100)//checking hundered'th digit for n>100
{
r=n/100;
n=n%100;
r--;
cout<<na1[r]<<" hundred ";
}
if(n>=20)//checking last 2 digits for n>20
{
r=n/10;
n=n%10;
r-=2;
cout<<" "<<na2[r];
}
if(n>=10)//checking unit digit for n>10
{
r=n%10;
n=0;
r--;
cout<<na3[r];
}
if(n>0)
cout<<" "<<na1[(n-1)];
getch();
}

/*   OUTPUT
Enter number:1254
one thousand two hundred  fifty four
*/

To be updated soon......
Gaurav Nigam

Read More