Conditional Statements Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

This discussion is designed to start you thinking about conditional statements, i.e., statements of the form “if something then something else.” I start by giving you some examples to build comfort with the conventions for when such a statement is true and when it’s false, and then some questions to start you thinking about how you might prove a conditional statement to be true.

Truth and Falsehood

Are the following conditional statements true or false? Why?

1. If x is a real number, then x+1 is a real number.
2. If 1.5 is a real number, then 2.5 is a real number.
3. If 1.5 is a real number, then 2.5 is an integer.
4. If x is a real number, then 3 is an integer.
5. If 1.5 is an integer, then pigs fly.

Suppose I tell you that “if π² is rational, then it’s raining gold coins” is a true statement. You look out the window and see that it is not raining gold coins. What can you conclude about π²? What if when you look out the window it is raining gold coins?

Proofs

Consider these two statements, one of which says, in a plain English version, that you can always fit a third real number in between any two other real numbers, and the second of which makes a similar claim about integers:

1. If x and y are real numbers such that x < y, then there is another real number, c, such that x < c < y.
2. If x and y are integers such that x < y, then there is another integer, c, such that x < c < y.

The first of these statements is true, but the second isn’t.

In terms of the rules for when a conditional statement is true and when it is false, what do you have to do to show that the second of my statements actually is false? (Our textbook touches on this, so consult it if you wish.)

Show how to do, or maybe start doing, that thing for the specific example here.

In terms of the rules for when a conditional statement is true and when it is false, what do you have to do to show that the first of my statements is in fact true?

Show how to do, or maybe start doing, that for the specific example here.