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Truth Table Examples

By David A. Smith

In this topic we illustrate the process of building truth tables. In terms of a truth table, statements p and q are called variables By definition, a proposition whose value is true for all cases of all variables is called a tautology; a proposition whose value is false for all cases of all variables is called a contradiction; and a proposition whose value depends on a value of a variable is called a contingency. Note that all four operations, And, Or, Not, and Implication are all contingencies. It is interesting to also note that we can take a combination of contingency statements and build a contradiction or a tautology. Recall that the contrapositive of is the statement and the converse of is the statement
    In this topic we will construct the truth tables for three tautologies, three contradictions, and three contingencies.

Example (Truth Table for a Tautology) Construct a truth table for the following statements.

(a) Tautology:  
    


(b) Tautology:  
    


(c) Tautology:  
    

Example (Truth Table for a Contradiction) Construct a truth table for the following statements.

(a) Contradiction:  
    


(b) Contradiction:  
    


(c) Contradiction:  
    

Example (Truth Table for a Contingency) Construct a truth table for the following statements.

(a) Contingency:  
    


(b) Contingency:  
    


(c) Contingency:  
    

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