Real Analysis

[DutifulJaguar9]

This is my first week in the class Real Analysis. I have been working through my homework and came across this problem:
9 Indicate whether each statement is true or false
(g) If 8 is even and 5 is not prime, then 4 < 7.
I thought false because the first half is false. The textbook says true. Can someone help with this, please?

 

[HallsofIvy]

You might want to review how your textbook defines "true" and "false" statements. The standard definition for "if p then q" is that such a statement is "false" only if p is true and q is false. In all other cases, in particular whenever p is false, the whole statement is true.

 

[pka]

This is my first week in the class Real Analysis. I have been working through my homework and came across this problem:
9 Indicate whether each statement is true or false
(g) If 8 is even and 5 is not prime, then 4 < 7.
I thought false because the first half is false. The textbook says true. Can someone help with this, please?

You should know that the basic connective are defined by truth tables: SEE HERE
Note that there is only one false case: \(\displaystyle T\implies F\)
It is said that:
A false statement implies any statement and a true statement is implied by any statement

 

[Dr.Peterson]

This is my first week in the class Real Analysis. I have been working through my homework and came across this problem:
9 Indicate whether each statement is true or false
(g) If 8 is even and 5 is not prime, then 4 < 7.
I thought false because the first half is false. The textbook says true. Can someone help with this, please?

This exercise is presumably intended to get you accustomed to concepts of mathematical logic, some of which are different from everyday use. Presumably the text explained this to you. Be sure to read the material preceding the exercise!

A statement "if p then q" says nothing about the case when p is false; it only makes a claim about the case when p is true. So if p is false, the statement is not a lie! You could say that a statement is "innocent until proven guilty" -- that is, in the absence of evidence that it is false (i.e. an example where p is true but q is not), we consider it true.

In ordinary usage, the most we might say is that we can't tell whether such a statement is true. But in mathematical logic, we have to call any statement either true or false, so the appropriate answer is that it is true. (In particular, this definition of a conditional statement is the only one that allows other rules about logic to be consistent.) This is what they are teaching you!

Addendum: I searched and found this exercise in a chapter from a textbook here; this book briefly states the definition and says more or less what I just said (bottom of page 4, top of page 5).

 

[DutifulJaguar9]

You might want to review how your textbook defines "true" and "false" statements. The standard definition for "if p then q" is that such a statement is "false" only if p is true and q is false. In all other cases, in particular whenever p is false, the whole statement is true.

Thank you. That helps a lot. I have always had trouble since high school with logic.

 

[HallsofIvy]

I've always thought of this as "innocent until proven guilty" (true until proven false). "p implies q" means "If p is true then q is true". If p is not true that statement says nothing about whether q is true or false.
But we need some value assigned to the statement- "true" is the default value.

 

[Steven G]

Here is how I think of it. If I have a million dollars, then you will get an A on your logic test. For this statement to be false you must show me that I have a million dollars and you did NOT get an A on your logic test. Good luck showing that I have a million dollars since I don't. So you can NEVER show that this if-then statement is false. So we call it true.

 

[Dr.Peterson]

In particular, this definition of a conditional statement is the only one that allows other rules about logic to be consistent.

I'll expand this comment I made previously:

A theorem is supposed to be a true statement. A proof has to demonstrate that the theorem is true.

Most theorems have the form, "If (conditions), then (conclusion)."

In order to say that such a theorem is proved to be always true, we want that conditional statement to be true, even when the conditions don't apply. This, ultimately, is why we define a conditional statement as being true in all cases except when the conditions are true but the conclusion is not. Only that definition of the truth value of a conditional statement allows theorems to called true!

(The same can be said, more strongly, about logical arguments, which are proved valid by using formal logic to show that the conditional statement expressing the argument is a tautology.)

I find this explanation to be stronger than just declaring that truth is the default, or that a statement is innocent until proved guilty (though those are both great ways to make the idea memorable).

 

[Otis]

If I have a million dollars, then you will get an A on your logic test.

I learned a couple months ago that this is true at several top universities in the USA.