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Are you referring f as the Image (Im f), or the inverse of f?
Since f is a function, for every element a∈A, there exists a b∈B, such that f(a) = b. Since f is injective, this b is unique.
Thus, |A| ≤ |B|.
Is that all I need to say for C to prove that part? Can I just say its true because the function is injective?
I understand the ideas of why its true, I'm just lost on how to contruct a formal proof for it. Any ideas?
Yes I mean the number of elements in the set
I need to prove the following and I'm completely lost.. Any help with be much appreciated.
Let A and B be finite sets and let f: A -> B be a function.
A) Prove that if f is injective, then |Im f| = |A|.
b) Is the converse of part (a) true? Prove or disprove.
c) Prove that if f is injective, then |A| ≤ |B|
Im f is the Image of f which i'm pretty sure just means the range.. This is the first time I've ever seen that used instead of the range of f. I think when you prove equality you need to show that whatever is on the left and right of the = needs to be shown as subsets of each other but I have no idea how to go about that.
I need to prove or disprove the following:
Let P(n,m) be the open sentence "n and m are odd integers" and Q(n,m) be the open sentence "nm is an odd integer".
a) For all integers n and m, P(n,m) -> Q(n,m) **-> means implies**
b) For all integers n and m, Q(n,m) -> P(n,m)
c) For all integers n and m, P(n,m) <-> Q(n,m)
I'm not sure how to go about proving these, any help would be much appreciated.