Math A

Truth Value of
Open  Sentences

 

Sometimes it is difficult to determine the truth value of a sentence.  Even though the sentence conveys a complete thought, the sentence may be true for some people and false for others.

For example:

  • "Broccoli tastes awful."

  • "The Beatles were an awesome group."

  • "Football is the most exciting sport to watch."


An even worse situation is the case where it is impossible to determine the truth value of a sentence due to a lack of information. 

For example:

  • "She did her homework."

  • "x + 5 = 25"

  • "It's the best movie this year."

Sentences that lack information are called ...

Open
Sentences!

  


An open sentence is a sentence which contains a variable.

  • "She did her homework." is an open sentence -- the variable is "She."

  • "x + 5 = 25" is an open sentence -- the variable is "x."

  • "It's the best movie this year." is an open sentence-- the variable is "It."

 

A variable is simply a spot waiting for a value.  The values we put into the variable are called the domain, or replacement set (because they "replace" the variable.)  The set of values which make the sentence TRUE is called the solution set, or truth set. 

Example:

Open sentence:  x + 5 = 25

Variable:  x

Domain:  {10, 20, 22, 24} 
   (numbers you can choose from)

  Solution Set:  {20}
 (the answer which makes
 the open sentence true)


Example:
 

Open sentence:
She did her homework.

Variable:  She

Domain:  {Sue, Melissa, Jennifer, Sandy, Joanne} 
(girls' names you can choose from)

Solution Set:
{Sue, Sandy}
(the answers which makes the open sentence true)
 (You would have to know which girls DID their homework.  In this case, Sue and Sandy did their homework.)

 

Remember:  Open sentences require that you have additional information to determine whether they are true or false.

 

   

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Roberts