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Section 1.1 Definitions

Definition 1.1.1.

A statement is a sentence which is either true or false. We will notationally speak of a statement \(p\) or a statement \(q\text{.}\)

Each of the following sentences are statements.

  1. George Washington was the first President of the United States.
  2. \(2 + 3 = 5\text{.}\)
  3. There are \(12\) inches in a foot.
  4. Harry S. Truman was the second President of the United States.
  5. \(4 \cdot 8 = 12\text{.}\)
  6. There are \(30\) inches in a yard.

Certainly we recognize some underlying knowledge is required in interpreting the sentences in Example 1.1.2. For instance, in the sentence “\(2 + 3 = 5\text{,}\)” we assume recognition of the concepts of \(2\text{,}\) \(3\text{,}\) \(5\text{,}\) and addition base \(10\text{.}\) Nevertheless, we must make some general knowledge assumptions, and certainly the six sentences of Example 1.1.2 are statements. (The first three sentences are true while the last three are false.)

Each of the following sentences are not statements.

  1. \(\displaystyle 2 + 3\)
  2. Study mathematics.
  3. Three is a nice number.
  4. Chris Hemsworth, who plays Thor in the Avengers series, is a handsome man.

Clearly, the problem in each of these sentences is that their truth or falsity cannot be uniquely determined. Actually, Item 3 and Item 4 could be statements provided that we have good definitions of “nice numbers” and “handsome”.

Consider the following sentence. “This sentence is false.” This is not a statement. Why? The sentence in question is an example of what is known in mathematics as a paradox. If it is true, then it is false, while on the other hand, if it is false, then it is true. Such paradoxes are not statements.

What, then, is an axiom? Surely it must be a statement, but also something more. As we study a body of mathematical knowledge, we encounter new statements, some of which can be proven from the existing system. These statements are called lemmas, theorems, facts, etc. Other statements cannot be proven. If we can in fact prove that the truth value of the statement is independent of the existing system, we have a potential axiom. We can either assume the statement is true, adding it to our system as an axiom, or we could assume the statement is false, adding its negative (or some form of its negative) to our system as an axiom. Surely, quite different bodies of knowledge would evolve depending on what axiom we added. The best examples of this concept are Euclidean geometry and the various non-Euclidean geometries.

Logicians as well as some other mathematicians are deeply concerned with such questions of creating minimal systems of axioms and developing mathematics very systematically from them. Such important but esoteric questions are beyond the scope and intent of this course. However, the remainder of this chapter will attempt to create a firm, logical base which will be used throughout this text and many subsequent courses the student will encounter.

For our purposes the student needs a basic feel for the flow of mathematics and a concrete understanding of the concept of bivalent logic applied to statements.

Exercises 1.1.1 Exercises

Determine whether or not the following sentences represent statements. If so, state the truth value.

1.

\(7 \cdot 9 = 63\text{.}\)

2.

There are more males than females registered in this class.

3.

Gone with the Wind is a good book.

4.

Eggs are a good source of calcium.

5.

\(64 \div 2 = 37\text{.}\)

6.

\(ax^2 + bx + c\text{.}\)

7.

\(ax^2 + bx + c = 0\text{.}\)

8.

The metric system of measurement is difficult to learn.

9.

Summer is the best season of the year.

10.

There are \(30\) people registered for this class.

11.

\(\sqrt{64} = 9\text{.}\)

12.

Today is a beautiful day.