3.3 Truth Tables for Negation, Conjunction, and Disjunction

Truth Values of Conjunctions and Disjunctions

Conjunctions are only true when both statements are true.

If M represents the statement “The moon is green with  yellow spots” and Y represents the statement “The sun is hot”, the statement "The moon is green with yellow spots and the sun is hot" may be written M Λ Y.  Since "The moon is green with yellow spots" is always false, this statement  M Λ Y is false.

Disjunctions are true if either of the statements is true.

If M represents the statement “The moon is green with  yellow spots” and Y represents the statement “The sun is hot”, the statement "The moon is green with yellow spots or the sun is hot"  may be written M V Y.  Since "The sun is hot" is always true, this statement  M V Y is true, even though one component was false.

 

In the previous two examples, the components of the conjunction and disjunction were known to be true or false. For example, we know that "the sun is hot" is always a true statement.  In general, we must account for all possible truth values. We may construct what is called a “truth” table to summarize all possible truth values.

Example: We may summarize all possibilities for the truth value of the statement P Λ  Q with the truth table shown below.

P Q P  Λ  Q
T T T
T F F
F T F
F F F

Note that every combination of truth values  is given in the left two columns and the resulting truth value is given in the right-most column. P  Λ  Q is only true when both P and Q are true.

 

Example: We may summarize all possibilities for the truth value of the statement PV Q with the truth table shown below.

P Q P  V  Q
T T T
T F T
F T T
F F F

Note that every combination of truth values  is given in the left two columns and the resulting truth value is given in the right-most column. P  V  Q is true when either P or Q are true.

Truth Values of Conjunctions and Disjunctions Combined With Negations

Example: We may summarize all possibilities for the truth value of the statement  ~P V ~Q with the truth table shown below.

P Q ~P ~Q ~P  V  ~Q
T T F F F
T F F T T
F T T F T
F F T T T

Note that we must construct additional columns for ~P and ~Q.  Then again, every combination of truth values  is given in the left two columns and the resulting truth values are given in the 3 right-most columns. The truth value of the negation of a statement is always the opposite as the truth value of the un-negated statement. ~P  V  ~Q is true when either ~P or ~Q are true.

Constructing Truth Tables

To construct a truth table containing two variables, use the following procedure:

  1. There are two columns with 4 rows underneath with the 4 possibilities of True and False. These two columns are at the far left.
  2. There are additional columns for negations of variables if needed.  The negations will have the opposite truth values.
  3. There are additional columns for each major component of the final statement.
  4. The last column at the right contains the final statement.

Example:  Construct a truth table for the statement  (P V ~Q) Λ ~P

1. Construct a table that has two columns for P & Q with the 4 possibilities for T & F.
 

P Q        
T T        
T F        
F T        
F F        

2. Make columns for ~P and ~Q

P Q ~P ~Q    
T T F F    
T F F T    
F T T F    
F F T T    

3. Make a column for P V ~Q

P Q ~P ~Q P V ~Q  
T T F F T  
T F F T T  
F T T F F  
F F T T T  

4. Make a column for (P V ~Q)  Λ  ~P

P Q ~P ~Q P V ~Q (P V ~Q)  Λ  ~P
T T F F T F
T F F T T F
F T T F F F
F F T T T T

HERE IS AN EXAMPLE OF A TRUTH TABLE FOR ~Q=>(P^Q)

DeMorgan's Law

If you construct a truth table for ~(P  Λ  Q) and a truth table for ~P V ~Q you can see that the resulting truth values on the far right are exactly the same for the same set of truth values for P & Q.

P Q ~P ~Q ~P V ~Q
T T F F F
T F F T T
F T T F T
F F T T T

 

P Q PΛ Q ~(P Λ Q)
T T T F
T F F T
F T F T
F F F T

We say that ~P V ~Q  is logically equivalent to ~(P Λ Q). 

Whenever two statements have the same truth values in the far right column for the same starting values of the variables within the statement we say the statements are logically equivalent.

De Morgan's Laws

De Morgan's Laws are two special cases of logical equivalence.  De Morgan's Laws state:

~P V ~Q   is logically equivalent to   ~(P Λ Q)

~P  Λ  ~Q   is logically equivalent to   ~(P V Q)

 


Assignment  -  Do the following problems of Section 3.3 of the text:  2,4,8,10,14,18,20, 22, 30,34,36
To create the tables, try copying and pasting the blank table shown below and editing it in your word processing program. This works on MSWord.

Copy From Here

P Q                
T T                
T F                
F T                
F F                

Through Here

If that does not work, simply line up your table values the best you can as shown in the example below

P  Q   ~P   ~PVQ
T  T     F       T
T  F     F       F
F  T     T       T
F  F     T       T 

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