Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

1.1. Statements and Conditional Statements 5
Progress Check 1.2 (Explorations)
Use the techniques of exploration to investigate each of the following statements.
Can you make a conjecture as to whether the statement is true or false? Can you
determine whether it is true or false?
1. .a C b/ 2 D a 2 C b 2 , for all real numbers a and b.
2. There are integers x and y such that 2x C 5y D 41.
3. If x is an even integer, then x 2 is an even integer.
4. If x and y are odd integers, then x y is an odd integer.
Conditional Statements
One of the most frequently used types of statements in mathematics is the so-called
conditional statement. 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. Since
conditional statements are probably the most important type of statement in mathematics,
we give a more formal definition.
Definition. A conditional statement is a statement that can be written in
the form “If P then Q,” where P and Q are sentences. For this conditional
statement, P is called the hypothesis and Q is called the conclusion.
Intuitively, “If P then Q” means that Q must be true whenever P is true.
Because conditional statements are used so often, a symbolic shorthand notation is
used to represent the conditional statement “If P then Q.” We will use the notation
P ! Q to represent “If P then Q.” When P and Q are statements, it seems
reasonable that the truth value (true or false) of the conditional statement P ! Q
depends on the truth values of P and Q. There are four cases to consider:
P is true and Q is true.
P is true and Q is false.
P is false and Q is true.
P is false and Q is false.