Question

Hello, i'm making a C++ program that creates a truth table for these statements:

1. ? ↔ ? ≡ ((? ∨ ?) ∧ (¬? ∨ ¬?))

2. ¬(? ∧ ?) ≡ ¬? ∨ ¬?

3. ? ↔ ? ≡ ((? → ?) ∧ (? → ?))

4. (? → ?) → ? ≡ ? → (? → ?)

I cannot use C++ standard library, and i'll be implementing the program using a linux command line.

for the math symbols i'll using the following:

?           P           M           P            M           P

∧           *            ∨         +                ¬          ~

⨁           ^            →          ->           ↔       ≡

=          ==

Here's a sample output:

Sample Output:

Equivalence: p->q === ~p+q

p             q             p->q       <->         ~p+q

0             0             1             1             1

0             1             1           1             1

1            0            0        1           0

1            1             1           1          1

p->q is equivalent to ~p+q

Here's the program I have so far, i'm not sure sure if its entirely correct. Please make any corrections and/or comments if needed Thank you!

#include<iostream>

using namespace std;

void method1(int x,int y)

{

// p <-> q == ((p v q) ^ (~pv~q))

cout<<"\n\n"<<x<<"\t"<<y<<"\t"<<((x==y)?"1":"0")<<"\t\t"<<((x|y)&(!x|!y));

}

void method2(int x,int y)

{

// ~(p^q) == ~p v ~q

cout<<"\n\n"<<x<<"\t"<<y<<"\t"<<!(x&y)<<"\t\t"<<(!x|!y);

}

void method3(int x,int y)

{

//p <-> q == ((p->q ^ (q->p)))

cout<<"\n\n"<<x<<"\t"<<y<<"\t"<<((x==y)?"1":"0")<<"\t\t"<<(((x==1)&&(y==0)?0:1)&((x==0)&&(y==1))?1:0);

}

void method4(int x,int y, int z)

{

// (p->q)-> r == p ->(q->r)

cout<<"\n\n"<<x<<"\t"<<y<<"\t"<<

(( (x==1) && (y==0) ?0:1) && (z==1)?1:0)<<"\t\t"<<((x==1)&&((y==1)&&(z==0)?1:0)?1:0);

}

int main()

{

int x,y,z;

cout<<"p\tq\t p<->q \t ((p v q) ^ (!p v !q)) \t";

for(x=0;x<=1;++x)

for(y=0;y<=1;++y)

{

method1(x,y);

}

cout<<"\nTable for Question 1: It is not valid as LHS not equal to RHS.\n";

cout<<"\np\tq \t!(p ^ q) \t (!p v !q) \t";

for(x=0;x<=1;++x)

for(y=0;y<=1;++y)

{

method2(x,y);

}

cout<<"\nTable for Question 2: It is valid as LHS is equal to RHS.\n";

cout<<"\np\tq\t (p <-> q) \t ((p->q) ^ (q->p)) \t";

for(x=0;x<=1;++x)

for(y=0;y<=1;++y)

{

method3(x,y);

}

cout<<"\nTable for Question 3: It is not valid as LHS not equal to RHS.\n";

cout<<"\np\tq\t (p -> q) -> r \t ((p->(q ->r)) \t";

for(x=0;x<=1;++x)

for(y=0;y<=1;++y)

for(z = 0; z<=1; ++z)

{

method4(x,y, z);

}

cout<<"\nTable for Question 4: It is not valid as LHS not equal to RHS.\n";

return 0;

}

Answer 1

The code seems to be almost correct. However, there are few things that are wrong.

1. method3() RHS calculation is wrong

Given : ((? → ?) ∧ (? → ?))

Your calculation: (((x==1)&&(y==0)?0:1)&((x==0)&&(y==1))?1:0)

You have mistakenly interchanged 1 and 0 at the very end of expression

Instead of doing it this way, you can do this:

Simplified expression : ((!? | ?) ∧ (!? | ?))

Therefore, question 3 is a valid expression (your ans was invalid)

There's some other thing I would like to point:

• Since you are dynamically generating truth table, you can not directly print the final result (valid or invalid) manually. This must be printed dynamically based on truth table. I have included that in code.

CODE:

#include<iostream>
using namespace std;

bool method1(int x, int y)
{
   // p <-> q == ((p v q) ^ (~pv~q))

   int lhs = int (x == y);
   int rhs = int ((x | y) & (!x | !y)); // this is equivalent to x != y

   cout << x << "\t" << y << "\t" << lhs << "\t\t" << rhs << "\n\n";
  
   return lhs == rhs;
}

bool method2(int x, int y)
{
   // ~(p^q) == ~p v ~q\
  
   int lhs = !(x & y);
   int rhs = (!x | !y);

   cout << x << "\t" << y << "\t" << lhs << "\t\t" << rhs << "\n\n";

   return lhs == rhs;
}

bool method3(int x, int y)
{
   //p <-> q == ((p->q ^ (q->p)))

   int lhs = int (x == y);
   int rhs = (!x | y) & (!y | x); // since p->q == ~p v q == ~p & q

   cout << x << "\t" << y << "\t" << lhs << "\t\t" << rhs << "\n\n";

   return lhs == rhs;
}

bool method4(int x, int y, int z)
{
   // (p->q)-> r == p ->(q->r)

   int lhs = (!(!x | y) | z);
   int rhs = (!x | (!y | z));

   cout << x << "\t" << y << "\t" << lhs << "\t\t" << rhs << "\n\n";

   return lhs == rhs;
}

int main()
{
   int x, y, z;
   bool isValid;
   string validStr = "It is valid as LHS equals to RHS";
   string invalidStr = "It is not valid as LHS not equal to RHS";

   cout << "Question 1 ===================\n";
   cout << "p"<<"\t"<<"q"<<"\t"<<"p<->q"<<"\t"<<"((p v q) ^ (!p v !q))"<<"\n\n";

   isValid = true;
   for (x = 0; x <= 1; ++x)
   for (y = 0; y <= 1; ++y)
           if (!method1(x, y))
               isValid = false;

   cout << (isValid ? validStr : invalidStr) << "\n\n";

   cout << "Question 2 ===================\n";
   cout << "p"<<"\t"<<"q"<<"\t"<<"!(p ^ q)"<<"\t"<<"(!p v !q)"<<"\n\n";

   isValid = true;
   for (x = 0; x <= 1; ++x)
   for (y = 0; y <= 1; ++y)
           if (!method2(x, y))
               isValid = false;

   cout << (isValid ? validStr : invalidStr) << "\n\n";

   cout << "Question 3 ===================\n";
   cout << "p"<<"\t"<<"q"<<"\t"<<"(p <-> q)"<<"\t"<<"((p->q) ^ (q->p))"<<"\n\n";

   isValid = true;
   for (x = 0; x <= 1; ++x)
   for (y = 0; y <= 1; ++y)
           if (!method3(x, y))
               isValid = false;

   cout << (isValid ? validStr : invalidStr) << "\n\n";

   cout << "Question 4 ===================\n";
   cout << "p"<<"\t"<<"q"<<"\t"<<"(p -> q) -> r"<<"\t"<<"(p->(q ->r)"<<"\n\n";

   isValid = true;
   for (x = 0; x <= 1; ++x)
   for (y = 0; y <= 1; ++y)
       for (z = 0; z <= 1; ++z)
               if (!method4(x, y, z))
                   isValid = false;

   cout << (isValid ? validStr : invalidStr) << "\n\n";

   return 0;

}

Screenshot:

Best wishes!