Question

1. Determine whether the following arguments are valid. Explain your answer.
   1. If it is hot outside, I will not go to play.
      If I play, I will not study.
      If I study, I will do good in the exam.
      I did good in the exam.
      Therefore, it is hot outside.
      

2. If I can run faster than air plane or I can kill a lion then I am a super human.
   If I am a super human then I am very careful about my diet.
   I am careless about my diet.
   Therefore, I cannot kill a lion.

2. How many times the “hello world” will be printed in the following algorithm if n =100:  

int Function(int n)
{
      for(int i = 0; i < n; i++)

      {

                  for(int j = 0; j < n; j++)

                  {

                              for(int k = 0; k < n; k++)

                              {

Print(“hello world”);

}

}

}

}

Answer 1

1.

Let the following symbols denote the given statements :

h : it is hot outside

p : I will go to play

s : I will study

e : I will do good in 'exam'

The given premises are :

• h -> ~p
• p -> ~s
• s -> e
• e

The conclusion to be proved is :

h

Let R be ( ( h -> ~p ) ∧ ( p -> ~s )  ∧ ( s -> e ) ∧ e ).

We need to show the truth value of ( ( h -> ~p ) ∧ ( p -> ~s )  ∧ ( s -> e ) ∧ e ) -> h

Let's use truth table for the given argument :

h	p	s	e	( h -> ~p )	( p -> ~s )	( s -> e )	R	R -> h
0	0	0	0	1	1	1	0	1
0	0	0	1	1	1	1	1	0
0	0	1	0	1	1	0	0	1
0	0	1	1	1	1	1	1	0
0	1	0	0	1	1	1	0	1
0	1	0	1	1	1	1	1	0
0	1	1	0	1	0	0	0	1
0	1	1	1	1	0	1	0	1
1	0	0	0	1	1	1	0	1
1	0	0	1	1	1	1	1	1
1	0	1	0	1	1	0	0	1
1	0	1	1	1	1	1	1	1
1	1	0	0	0	1	1	0	1
1	1	0	1	0	1	1	0	1
1	1	1	0	0	0	0	0	1
1	1	1	1	0	0	1	0	1

The final output that is R -> h or in other words, ( ( h -> ~p ) ∧ ( p -> ~s )  ∧ ( s -> e ) ∧ e ) -> h is not a tautology.

Hence, the given argument is not valid.

2.

Let the following symbols denote the given statements :

a : I can run faster than air plane

l : I can kill a lion

s :  I am a super human

d : I am very careful about my diet

The given premises are :

• ( a v l ) -> s
• s -> d
• ~d

The conclusion to be proved is :

~l

The given argument can be proved using the following table :

Statement no.	Statement	Justification
1	( a v l ) -> s	given
2	s -> d	given
3	~d	given
4	~s	Modus tollens rule on statements 2 and 3
5	~(a v l )	Modus tollens rule on statements 1 and 4
6	~ a ∧ ~ l	De Morgan's Law on statement 5
7	~l	Simplification rule on statement 6

The conclusion, ~l, can be derived from the given premises.

Hence, the given argument is valid.

3.

In the function, there are 3 for loops which are independent of each other and runs for O(n) times each.

So, total number of times the function executes is O(n^3) times.

Here, n = 100.

Then, the number of times the statement “hello world” will be printed is ( 100 * 100 * 100 ) = 1000000 times.

Hence, the answer is 1000000 times.