1. Construct a truth table for 

p	q	r					
T	T	T	F	T	T	T	
T	T	F	F	T	T	T	
T	F	T	F	F	T	T	
T	F	F	F	F	T	T	
F	T	T	T	T	T	T	
F	T	F	T	T	F	F	
F	F	T	T	T	T	T	
F	F	F	T	T	F	F	

2. Give a truth table for 

T	T	F	T	T	T	
T	F	F	F	F	T	
F	T	T	T	T	T	
F	F	T	T	T	T	

3. Consider the sentence “All dragons lay eggs.”  Circle all of the sentences below which are negations of this sentence.   (There may be more than one answer.)

i.  No dragons lay eggs.			ii.  Not all dragons lay eggs.	

iii.  Some dragon lays eggs. 		iv.  Some dragon doesn’t lay eggs  

4. Decide if the argument is valid or invalid.  Use proof by contradiction to prove your assertion.

                 

Proof by contradiction: Assume that the conclusion is false. This means the statement s is false.
In order for the premiseto be true,  must be true since s is already false.
In order for the premise to be true,  could be true or false since  is already true. Assume r is false.
In order for the premise to be true, both and  need to be true. Since is already false, is true. If we pick t to be true, the first premise is true. We were able to make all the premises true, therefore there is no contradiction, so the argument is invalid. 

5. Prove that the following argument is valid using the Rules of Inference.

        

Proof by using the rules of inference:
    (Disjunctive Syllogism);            (Modus tollens);           (Disjunctive Syllogism)

6. Consider the conditional statement:  “If Pinocchio tells a lie, then either his nose will grow or he will turn into a frog.”  For this conditional statement, give:

a. The negation.
Let  be the statement: Pinocchio tells a lie; let  be the statement: His nose grows; and let  be the statement: He turns into a frog. Thus, the conditional statement can be described as:

Therefore, the negation of the statement is: 

which is: Pinocchio tells a lie and his nose will not grow and he will not turn into a frog.

or: Pinocchio tells a lie and neither his nose will grow nor will he turn into a frog. 

b. The converse.
The converse of is 
which is: If Pinocchio’s nose grows or he turns into a frog, then he will tell a lie.

7. Use a circle diagram (Venn diagram) to decide if the following arguments are valid or invalid.

a. Some lizards are vegetarian.  
   All vegetarians do yoga.
   Iggy does yoga.
Iggy is a lizard.

From the Venn diagram:
Iggy doesn’t need to be a lizard.
	Therefore, the argument is invalid.

Circle your conclusion:  VALID		INVALID

b.  	All frogs are green.  
	No one who is green can sing.
	Kermit can sing.
	Kermit is not a frog.

Circle your conclusion:  VALID		INVALID

	From the Venn diagram:
	Kermit cannot be a frog
	Therefore, the argument is valid.

8.	Let p, q, r and s be the statements.
		p: My clothes are dirty.                           q: It’s time to do laundry.
		r: All of the laundromats are closed.       s: Some of the laundromats are closed.

Use the logical connective symbols and the letters p, q, and r, to represent the following statements. 

a. If my clothes are dirty then it’s time to do laundry.
p  q

b. It’s time to do laundry if and only if all of the laundomats are closed.
 	q  r

c. My clothes are dirty, or, it’s time to do laundry and all of the laundromats are closed.
 	p  (q  r)

d. Some laundromats are open.
	r

e. All of the laundromats are open.
	s

9. Given the following argument: 
   	 	p  q
    		q  s
   		 s_____ 
     		p
	Show that it is valid using: 
a. a truth table 
	 
p	q	S	q	p  q	q  s	s	(p  q) 
( q  s )( s)	p	((p  q) ( q  s )
( s))  p	
T	T	T	F	F	T	F	F	F	T	
T	T	F	F	F	T	T	F	F	T	
T	F	T	T	T	T	F	F	F	T	
T	F	F	T	T	F	T	F	F	T	
F	T	T	F	T	T	F	F	T	T	
F	T	F	F	T	T	T	T	T	T	
F	F	T	T	T	T	F	F	T	T	
F	F	F	T	T	F	T	F	T	T	

b. the method of contradiction	

Assume p  q, q  s, and s are all true, and that p is false. Since p is false, p 
is true. Since p  q is also true, q must be true, so q is false. Since s is true, s 	is also false. This means q  s should be false, which contradicts the assumption that it’s true, so p can’t be false, so the argument is valid.

c. the rules of inference.

Using contraposition with double negation changes p  q to q  p giving:
		q  p                                                                                        
         	q  s                                               
   		 s_____                                                                         
     	p                                               
		Using disjunctive syl. q  s and s means q is true so we have:
		q  p
		q______
		p
		Which is true by modus ponens.     

10. Prove the following argument using the rules of inference.
         p  q
         r  q
         t  r
        (p  s)
          t

	Using contraposition, r  q becomes q r , giving:  p  q
         q r             
  	 t  r  
	 (p  s)
	  t
	
Using hypothetical syl. p  q and q r  means    p  r, so now we have:
	p  r
	t  r           
	(p  s)   
	  t
 	Using DeMorgan’s laws with double negation changes (p  s) to p  s , giving: 
   	p  r
	t  r
	p  s  
	 t
	Reordering and using simplification on p  s  gives:
	t  r                             
	p  r
	p______
	 t
	Modus ponens with p  r and p means r is true leaving:
	t  r       
	r____
    	 t      
	Which is true by disjunctive syllogism.

11.	Determine whether the following arguments are valid or not using a diagram.
    	a)	All elephants are large. 	                    b)  All elephants are large.
		Some pigs are not large. 			        Some pigs are large._____
          No pigs are elephants.	                                Some pigs are elephants.
		
          Invalid							invalid

   	c)	All elephants are large.
		Some pigs are not large.
          Some pigs are not elephants.
	valid
a,c.	b.