Truth Functions

• We have seen that an individual statement may be True or False.
  
  
• Each statement may contain one or more operators (i.e., if ... then, and, or, not).
  
  
• For each operator we can set up a Truth Function to show all the possible True/False values.
  
  
• For example, the simplest expression containing the operator and looks like this:
  
  

  Elements		Expression
  P	Q	P Q
  T	T	T
  T	F	F
  F	T	F
  F	F	F

  
  
  • Where we have assigned True and False values to each of the Elements (in this case the P and the Q).
    
    
  • If the value of P is True, and the value of Q is True, then the value of P and Q is True.
    
    
  • However, if the value of P is True and the value of Q is False, then the value of P and Q is False.
    
    
• Similarly, the simplest expression containing the operator or looks like this:
  
  

  Elements		Expression
  P	Q	P Q
  T	T	T
  T	F	T
  F	T	T
  F	F	F

  
  
  • Remember, the operator or really means at least one is the case! So, the only time the expression is False is when both P and Q are False.
    
    
• The truth function for the operator if ... then ... is a sometimes a little harder to grasp. It looks like this:
  
  

  Elements		Expression
  P	Q	P Q
  T	T	T
  T	F	F
  F	T	T
  F	F	T

  
  
  • You might interpret the truth function this way - Can the expression be True if certain elements are False?
    
    
  • For example, can the expression be True if P is False and Q is True? Yes, because the expression really tells us that when P is True Q must also be True. In fact, Q can be True whether P is True or False.
    
    
  • However, if Q is False and P is True, then the expression If P then Q cannot be True.
    
    
  • Finally, can the expression be True if both P and Q are False? Yes, because the expression really only tells us that when P is True Q must also be True.
    
    
• We will use these Truth Functions to make Truth Tables. Those Truth Tables will help us determine Validity.