Q. Write a function definition for a function void show_mid(int p[ ][ ], int r, int c) in C++ to display the elements of middle row and middle column from a two dimensional array p having number of columns r and number of rows c.

#include<conio.h>
#include<iostream.h>

/* FUNCTION DEFINITION TO SHOW ELEMENT OF MIDDLE ROW AND COLUMN */

void show_mid(int p['z']['z'],int r, int c)
{
int midi,midc;
      if(r%2==0||c%2==0)
      {
          cout<<"\nGiven array has no Middle Row/Column";
            return;
         }
midi = (r+1)/2;
midc = (c+1)/2;
cout<<"\nElements of Middle Row are :\n";
do
{
for(int j=0;j<c;j++)
{
cout<<p[midi][j]<<"\t";
}

}while(0);
cout<<"\nElements of Middle Column are :\n";
do
{
for(int j=0;j<r;j++)
{
cout<<p[j][midc]<<"\t";
}

}while(0);
}

/* MAIN FUNCTION */

void main()
{
int a['z']['z'],r,c;
cout<<"\nEnter No. of Rows : ";
cin>>r;
cout<<"\nEnter No. of Columns : ";
cin>>c;
cout<<"\nEnter an Array : \n";
for(int i=0;i<r;i++)
{
for(int j=0;j<c;j++)
{
cin>>a[i][j];
}
}
show_mid(a,r,c);
getch();

}

Basic Theorems of Boolean Algebra

Basic Theorem / Postulates of Boolean Algebra and its Truth Tables

1.       Properties of 0 and 1

1.1.    0 + X = X

0	X	R
0	0	0
0	1	1

1.2.    0.X = 0

0	X	R
0	0	0
0	1	0

1.3.    1 + X = 1

1	X	R
1	0	1
1	1	1

1.4.    1.X = X

1	X	R
1	0	0
1	1	1

2.       Idempotent Law

2.1.    X + X = X

X	X	R
0	0	0
1	1	1

2.2.    X.X = X          

     

X	X	R
1	0	0
1	1	1

3.       Involution

3.1.    (X’)’ = X

X	X’	(X’)’
0	1	0
1	0	1

4.       Complement Law

4.1.    X + X’ = 1

X	X’	R
0	1	1
1	0	1

4.2.    X.X’ = 0

X	X’	R
0	1	0
1	0	0

5.       Commutative Law

5.1.    X + Y = Y + X

X	Y	X+Y	Y+X
0	0	0	0
0	1	1	1
1	0	1	1
1	1	1	1

5.2.    X.Y = Y.X

X	Y	X.Y	Y.X
0	0	0	0
0	1	0	0
1	0	0	0
1	1	1	1

6.       Associative Law

6.1.    X + (Y + Z) = (X + Y) + Z

X	Y	Z	X+Y	Y+Z	X+(Y+Z)	(X+Y)+Z
0	0	0	0	0	0	0
0	0	1	0	1	1	1
0	1	0	1	1	1	1
0	1	1	1	1	1	1
1	0	0	1	0	1	1
1	0	1	1	1	1	1
1	1	0	1	1	1	1
1	1	1	1	1	1	1

  

6.2.    X.(Y.Z) = (X.Y).Z

X	Y	Z	X.Y	Y.Z	X.(Y.Z)	(X.Y).Z
0	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	1	1	0	1	0	0
1	0	0	0	0	0	0
1	0	1	0	0	0	0
1	1	0	1	0	0	0
1	1	1	1	1	1	1

7.       Distributive Law

7.1.    X(Y + Z) = XY + XZ

X	Y	Z	Y+Z	X.Y	X.Z	X.(Y+Z)	X.Y+X.Z
0	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	1	0	1	0	0	0	0
0	1	1	1	0	0	0	0
1	0	0	0	0	0	0	0
1	0	1	1	0	1	1	1
1	1	0	1	1	0	1	1
1	1	1	1	1	1	1	1

  7.2.    X + (Y.Z) = (X + Y ).(X + Z) 

X	Y	Z	Y.Z	X+Y	X+Z	X+(Y.Z)	(X+Y).(X+Z)
0	0	0	0	0	0	0	0
0	0	1	0	0	1	0	0
0	1	0	0	1	0	0	0
0	1	1	1	1	1	1	1
1	0	0	0	1	1	1	1
1	0	1	0	1	1	1	1
1	1	0	0	1	1	1	1
1	1	1	1	1	1	1	1

 8.       Absorption Law 

8.1.    X + XY = X

X	Y	X.Y	X+(X.Y)
0	0	0	0
0	1	0	0
1	0	0	1
1	1	1	1

  

8.2.    X .(X + Y) = X

X	Y	X+Y	X.(X+Y)
0	0	0	0
0	1	1	0
1	0	1	1
1	1	1	1

 9.       3^rd Distributive Law

9.1.    X + X’Y = X + Y

X	Y	X’	X’.Y	X+ X’Y	X+Y
0	0	1	0	0	0
0	1	1	1	1	1
1	0	0	0	1	1
1	1	0	0	1	1