Hello, all! I've spent the last 2 hours trying to do this question, and my paper is still blank!
Question:
For integers m and n, let A[m,n] = where x is an integer.
(a) List all the members of the set [-2,4]
(b) For , write down a formula in terms of m and n, for the size of the set A[m,n]. Justify your answer by giving an explicit , for an appropriate value of k.
My attempts:
(a) {-2, -1, 0, 1, 2, 3, 4}
(b) I'm thinking the formula will be n-m? ButI have no real idea, to be honest.
Please help. Thank you.
Thanks for the reply. I didn't even think about applying it to the set in part (a)! Of course, the set has 7 elements, and n-m is just 6.
Thanks to you too. So if , then we can define f to be the bijection .Originally Posted by plato
But I don't understand where the f(k) = k+m bit comes from. Furthermore, I don't see why it is useful. Of course, I'm not saying it isn't, it's just that I fail to see why - but I am hopeless at this stuff. The reason I'm saying this is because I can't see a way to find a numerical value for k, or don't you have to?
Thanks once again guys, more help will be appreciated greatly.
I see! Thank you
Can I be a bit cheeky and ask for help on another question? It's a very short one...
Question:
Give an example of an injection that is not a bijection. ( is meant to be the set of Natural Numbers, btw.)
My thinking:
My first inclination is that you can't.
By definition, an injection is where every y in Y is value f(x) for at most one x in X.
So would I be right in saying that an injection is such an injection. If I am right, how would I go about writing this in 'proper' notation?
An injection is a function that has values but there is not necessarily . In other words, we must be able to use all of A, but we don't necessarily use all of B.
So your example is correct: you are using all the natural numbers as your domain, and only the even natural numbers are used as your range. But to make this an injective function we can't use the set because then we'd have a one to one function. So an injective version of this function will be:
whereas the bijective version is
where is the set of even whole numbers.
-Dan
Thanks a lot Dan!
I understood what you said completely. I think (or hope) I understand the concept behind it, it's just the notation I'm not used to. I've only started covering this stuff at university, so it all seems a bit alien to me at the moment.
A huge thank you once again to you, Plato and Jhevon!
First, when asking a second, follow-up question it is best to start a new thread.
As to 'proper' notation, that is always defined by your textbook/instructor.
I see that Dan has given you a possible answer.
Here is another: .
In some foundation courses that is the preferred notation because it points to a function as a set of ordered pairs.