A. Statements

Do you ever answer your teacher’s question?  If so, how is the answer that you have given?  True or false?  If you answer a question completely, your answer may perform a sentence. For example, “Where do you study?” An example of the answer is “I study at SMPN 1 Wonosari”.
Now, study the following sentences.
(a) The number of players in a football team is 12.
(b) The number of players in a volleyball team is 6.
(c) 2 is not a prime number.
(d) Surabaya is the capital of East Java.
(e) 21 + 15 > 35
(f) –10 > -8
(g) A negative number plus a negative number is a positive number.
(e) A prime number is a number that has exactly two factors, i.e. 1 and the number itself.
Which sentences are true? Which sentences are false? If a sentence is false, give your reason why.

A true or false sentence is called a statement.

B. Open Sentences

Gift Problem
1. In her birthday party, Ani received some books as a gift from Johan.  Lisa said, “the number of the books in that gift-box is six pieces”.

What is your opinion about Lisa’s statement? True or false?

2. A fruit seller received a shipment of oranges in a sack, the number of which was unknown.  The number of the remaining oranges was 34.  The seller said, “the total number of the oranges is 456”.  Can you determine the truth-value (True or False) of the seller’s statement? Why?

3. Consider the sentence “A number added by 5 is 12.” Can you determine the truth-value of the sentence?

We cannot determine the truth-value of the sentence because the value of “a number” in that sentence has not been determined. Whether it is true or false depends on what “a number” is.

If “a number ” is 7, then the sentence becomes “7 added by 5 is 12”. The truth value of the sentence is true. If “a number ” is 10, then the sentence becomes “10 added by 5 is 12”. The truth value of the sentence is false.  If “a number ” is -4, then the sentence becomes “-4 added by  5 is 12”. The truth value of the sentence is false.

Therefore, the sentence “A number added by 5 is 12” is a sentence the truth value of which has not been determined yet. A sentence the truth value of which has not been determined is called an open sentence.

The value of “a number ” in the above sentence has not been determined yet. Mathematically, the value which has not been determined yet is called a variable, symbolized by a letter in lowercase, such  as x, y, m or others.

Therefore, the sentence “A number added by 5 is 12” can be symbolized as x + 5 = 12 , or y + 5 = 12 , or  m + 5 = 12.  Thus, the variable of the open sentence x + 5 = 12 is x. The variable of the open sentence  y + 5 = 12 is y, and the variable of the open sentence  m + 5 = 12 is m.