I have just sent the following message to my first-year students. But I am not sure whether they will find this helpful or confusing.
Since they have a Class Test later today, this may have been the wrong time to send it anyway!
Message sent to first-year students follows:
Looking through some more of your work on Practice Coursework 1, I have noticed one more logical reasoning point that has come up, related to negating statements (when trying to prove things by contradiction). This can be a bit tricky if you are trying to negate a statement which involves an implication sign or the word “if”. (In the solutions you will see that I have generally been negating statements of a more tractable type. I recommend that, if you can, you keep things simple when you are negating things!)
In order to prove true a statement of the form A implies B, some of you have said “Assume towards a contradiction that A implies that (B is false)” (or something like that). However, the correct negation of “A implies B” is instead “A does not imply B”.
One comment here is that it should not be possible for a statement and its negation to both be false. Nor should it be possible for them both to be true.
In the below, we assume that n is an integer, but make no other assumptions about n before considering whether certain logical implications are true or false.
Consider the false statement (about integers n) that says
“n is divisible by 3 implies that n is divisible by 6”.
(By false here we really mean “false for at least one integer n”. So consider n=3.)
Here the statement
“n is divisible by 3 implies that n is not divisible by 6″
is also false! (Again meaning false for at least one n. This time consider n=6.)
The correct negation of the first false statement is
“n is divisible by 3 does not imply that n is divisible by 6”
(by which, this time, we mean that there is at least one integer n such that n is divisible by 3 but n is not divisible by 6).
As you would expect, this correct negation of the first false statement is actually true.
Note that we have followed certain conventions here when discussing whether these implications are true or false. You may wish to rewrite the various claims above more formally using “For all integers n” and “There exists an integer n” to clarify things.
Using the mathematicians’ truth value for “implies”, you may wish to note some relatively strange things:
“4 is divisible by 3 implies that 4 is divisible by 6”
is (officially) true, and the statement
“4 is divisible by 3 implies that 4 is not divisible by 6″
is also (officially) true! So that can’t be the correct negation of the first statement.
“4 is divisible by 3 does not imply that 4 is divisible by 6”