Confusion about proving logical implication statements I've got four statements* which I'm meant to evaluate as being either true or false.
a. If 25 is a multiple of 5, then 30 is divisible by 10.
b. If 25 is a multiple of 4, then 30 is divisible by 10.
c. If 25 is a multiple of 5, then 30 is divisible by 7.
d. If 25 is a multiple of 4, then 30 is divisible by 7.
While I'm inclined to just say they're all false, since I can't find any way that the then statements follow from the if statements, I can't help but feel as if I'm missing something, in that we're meant to assume the if statements are correct.
The best I can guess is that by accepting the conditionals, the number system must be warped in some way through which the following then statements are analyzed.
Anyone able to provide any insight?
 A: Welcome to MSE.
You’re doing a classic problem in logic. The idea is that, from a false statement, you can derive any other statement.
Hence we have the truth table for the statement “if P, then Q”. We call this $S$:
$S = (P \Rightarrow Q)$
P false, Q true: S true
P false, Q false: S true
P true, Q true: S true
P true, Q false: S false
Can you answer your question now?
A: I think these problems are intended to test your understanding of material implication.
The idea is that when a mathematician uses the word "if", they usually don't really mean "if". They actually mean something completely different, and the word "if" is merely a convenient (but inaccurate and confusing) abbreviation for what they really mean.
When a mathematician writes "if A, then B", what they really mean is "either not-A, or B".
So what these statements actually mean is:
a. Either 25 is not a multiple of 5, or 30 is divisible by 10.
b. Either 25 is not a multiple of 4, or 30 is divisible by 10.
c. Either 25 is not a multiple of 5, or 30 is divisible by 7.
d. Either 25 is not a multiple of 4, or 30 is divisible by 7.
Just ignore the confusing usage of the word "if", rewrite the sentences to use the words "not" and "or", and continue on.
A: The statement
If $X$, then $Y$ is considered to be true when either $X$ and $Y$ are both true, or if $X$ is false. And so (b) and (d) are already true. Note that $25$ being a multiple of $5$ doesn’t imply that $30$ is a multiple of $7$, so (c) is false. For (a), both the ‘if’ and ‘then’ statements are true, and so (a) is true as well.
