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MAT 112 Integers and Modern Applications for the Uninitiated

Sebastian Pauli

Section 1.2 Statements

Statements are declarative sentences that are either true or false. The statements are formulated in such a way that any reader, who knows what all the words mean, can understand them.

Example 1.13. Non-mathematical statements.

  1. “Victoria likes cookies.” is a declarative sentence, and it is either true or false, so it is a statement.
  2. “Broccoli is green.” is a declarative sentence and it is true, so it is a statement.
  3. “Broccoli is pink.” is a declarative sentence and it is false, so it is a statement.
  4. “Cookies!” is not a declarative sentence, so it is not a statement.

Subsection Statements about integers

From now on we concentrate on statements about the integers.

Example 1.14. Statements about integers.

Consider the following:
  1. “2 is equal to 3.” is a statement. It is false.
  2. “2 plus 3 is equal to 5.” is a statement. It is true.
  3. “2 plus 3” is not a statement, as it is not a declarative sentence; it is not even a sentence, as it does not contain a verb.
When we write a statement using the symbols \(=\text{,}\) \(\ne\text{,}\) \(\lt\text{,}\) \(\le\text{,}\) \(>\text{,}\) or \(\ge\text{,}\) the comparison symbol takes the place of the verb. A mathematical statement always has a verb or a symbol that takes the place of the verb, just as a sentence does.
A mathematical expression consists of objects and operations. The objects can be numbers or variables (see the next section) and the operations can be, for example \(+\text{,}\) \(\cdot\text{,}\) or \(-\text{.}\) Unlike a statement, an expression has no comparison symbol, that means it has no “verb.” So expressions by themselves are not true or false, but expressions can be used in statements, as in Example 1.6.

Example 1.15. Statements with \(=\).

We formulate Example 1.14 using symbols.
  1. “2 = 3” is a statement. It is false.
  2. “2 + 3 = 5” is a statement. It is true.
  3. “2 + 3” is an expression. As it does not have a verb it is not a statement.

Example 1.16. True and false statements.

We identify whether statements about integers are true or false. Notice that all these examples when read out have a verb in them, namely “is”.
  1. \(2=2\) is read “2 is equal to 2”. This is a true statement.
  2. \(2=3\) is read “2 is equal to 3”. This is a false statement.
  3. \(2\gt 3\) is read “2 is greater than”. This is a false statement.
  4. \(2\ne 3\) is read “2 is not equal to 3”. This is a true statement.
  5. \(2\le 2\) is read “2 is less than or equal to 2”. This is a true statement.

Example 1.17. Expressions and statements.

We give some examples of expressions and statements and identify them.
  1. “\(2 + 3\)” is an expression.
  2. “\(2+3 = 5\)” is a statement.
  3. “\(2 + 1 + 5\)” is an expression.
  4. “\(2+ 1 + 5 \lt 10\)” is a statement.

Problem 1.18. Decide whether statements are true or false.

Decide whether the following are statements or not. If they are statements decide whether they are true or false.
  1. “Sunflower”
  2. “Stop signs are red.”
  3. “\(2\) is equal to \(3\text{.}\)”
  4. \(\displaystyle (1+2)-4687\)
  5. \(\displaystyle 2+3=7\)
  6. \(\displaystyle 3 > -100\)
Solution.
  1. “Sunflower” is not a sentence, so it is not a statement.
  2. “Stop signs are red.” is a declarative sentence, so it is a statement. It is true.
  3. “2 is equal to 3” is a declarative sentence, so it is a statement. As \(2\ne 3\) the statement is false.
  4. \((1+2)-4687\) is not a statement as it has no verb.
  5. \(2+3=7\) is a statement, the verb is ‘=’ (is equal to). As \(2+3=5\) it is a false statement.
  6. \(3 > -100\) is a statement, the verb is “\(>\)” (is greater than). It is a true statement.
When a statement is true, we usually do not write “is true.” When a statement is false, always write “is false”.
In Checkpoint 1.19 recognize statements and for statements decide whether they are true or false.

Checkpoint 1.19. Are these statements ? If yes, are they true or false.

Determine which of the following are mathematical statements.
For the statements decide whether they are true or false.
  1. \(\displaystyle 13\le 38\)
  2. \(\displaystyle -11-38\)
  3. \(\displaystyle 13\cdot 38\)
  4. \(\displaystyle 13\ne 38\)

Subsection Compound Statements

In mathematics we often deal with multiple statements that overlap. In these cases instead of writing each statement separately, we often write them as one string of statements. This allows us to connect the statements directly.

Example 1.20.

Instead of writing “\(2 + 3 = 5\)” and “\(5 = 1+4\text{,}\)” we write “\(2 + 3 = 5 = 1+4\text{.}\)”
We can also do this with inequalities.

Example 1.21.

Writing “\(2+5 = 7 \lt 10\)” means both “2+5 = 7” and “\(7 \lt 10\text{.}\)” In words, “\(2\) plus \(5\) is \(7\) and \(7\) is less than \(10\text{.}\)”
Compound statements are often used to prove identities, that is, when proving that two expressions are equal. The proofs of Theorem 1.59 and Theorem 1.60 and Theorem 1.65 in the next chapter is written that way.
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