Mathematical Reasoning – Writing and Proof

Chapter 1 – Introduction to Writing Proofs in Mathematics

1.1 Statements and Conditional Statements

Beginning Activity 1 (Statements)

Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that must have a definite truth value, either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as “The sky is beautiful” is not a statement since whether the sentence is true or not is a matter of opinion. A question such as “Is it raining?” is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2x C 5 D 10 is not a statement since we do not know what x represents. If we substitute a specific value for x (such as x D 3), then the resulting equation, 2 . 3 + 5 = 10 is a statement (which is a false statement).

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

1. 3 – 4 + 7 = 19.
2. 3 – 5 + 7 = 19.
3. 3x + 7 = 19.
4. There exists an integer x such that 3x + 7 = 19.
5. The derivative of
6. Does the equation  have two real number solutions?

Beginning Activity 2 (Conditional Statements)

Given statements P and Q, a statement of the form “If P then Q” is called a conditional statement. It seems reasonable that the truth value (true or false) of the conditional statement “If P then Q” depends on the truth values of P and Q. The statement “If P then Q” means that Q must be true whenever P is true. The statement P is called the hypothesis of the conditional statement, and the statement Q is called the conclusion of the conditional statement. We will now explore some examples.

1. “If it is raining, then Laura is at the theater.” Under what conditions is this conditional statement false? For example,(a) Is it false if it is raining and Laura is at the theater?
   (b) Is it false if it is raining and Laura is not at the theater?
   (c) Is it false if it is not raining and Laura is at the theater?
   (d) Is it false if it is not raining and Laura is not at the theater?
2. Identify the hypothesis and the conclusion for each of the following conditional statements.(a) If x is a positive real number, then is a positive real number.
   (b) If   is not a real number, then x is a negative real number.
   (c) If the lengths of the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.

Statements

As we saw in Beginning Activity 1, some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. Following are some more examples

• There exists a real number x such that x + 7 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely x = 3.
•  For each real number x, 2x + 5 = 2 (x + 5/2)
  This is a statement since either the sentence 2x + 5 = 2 (x + 5/2) is true when any real number is substituted for x (in which case, the statement is true) or there is at least one real number that can be substituted for x and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
• Solve the equation
  This is not a statement since it is a directive. It does not assert that something is true.
• is not a statement since it is not known what a and b represent. However, the sentence, “There exist real numbers a and b such that ” is a statement. In fact, this is a true statement since there are such integers. For example, if a D 1 and b D 0, then .
• Compare the statement in the previous item to the statement, “For all real numbers a and b, ” This is a false statement since there are values for a and b for which For example, if a = 2 and b =D 3, then

Progress Check 1.1 (Statements)

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

1. 2 – 7 + 8 = 2
2. 2. 2x + 5y = 7.
3. There are integers x and y such that 2x C 5y D 7:
4. Given a line L and a point P not on that line and in the same plane, there is a unique line in that plane through P that does not intersect L.
5. 
6. For all real numbers a and b, .
7. Does the equation have two real number solutions?
8. If ABC is a right triangle with right angle at vertex B , and if D is the midpoint of the hypotenuse, then the line segment connecting vertex B to D is half the length of the hypotenuse.
9. There do not exist three integers x, y, and z such that