Review Suggestions

•	Be able to write English sentences corresponding to statements formed using logical connectives and visa versa. 
•	Be able to correctly negate a quantifier. 
•	Be able to state the converse, the inverse and the contra-positive of a conditional statement 
•	Be able to construct a truth table of a statement
•	Be able to decide if an argument is valid using truth table, rules of inference or the method of contradiction.
•	Use Venn diagrams to decide the validity of an argument
You will be given the truth tables of the basic connectors and the list of inference rules 

Logical Connectors

p	p		p	q	pq		p	q	pq		p	q	pq		p	q	p q	
T	F		T	T	T		T	T	T		T	T	T		T	T	T	
F	T		T	F	F		T	F	T		T	F	F		T	F	F	
		F	T	F		F	T	T		F	T	T		F	T	F	
		F	F	F		F	F	F		F	F	T		F	F	T	

Rules of Inference

Double Negation		Implication		Simplification	
(P)	
P		    P	
(P)		P  Q
 P  Q	P  Q
P  Q		P  Q
P	P  Q
Q	
Modus Ponens		Modus Tollens		Contraposition	
 P  Q
 P	
Q		P  Q
Q	
P		     P  Q	
Q   P	
Hypothetical Syllogism		Disjunctive Syllogism		De Morgan’s Laws	
	P  Q
	Q  R	
	P  R		P  Q
P	
 Q		P  Q
Q	
 P			 (P  Q)	
 P  Q	 (P  Q)	
 P  Q	

Sample Problems

1. Construct a truth table for 

2. Give a truth table for 

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.

                 

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

        

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.

b. The converse.

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.

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

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.

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

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

d. Some laundromats are open.

e. All of the laundromats are open.

9. Given the following argument: 
   	 	p  q
    		q  s
   		 s_____ 
     		p
	Show that it is valid using: 

a. a truth table 

b. the method of contradiction	

c. the rules of inference.

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

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.

   	c)	All elephants are large.
		Some pigs are not large.
          Some pigs are not elephants.