Question

Let A and B be sets.Let x⊆A be a subset and let f:A --> B be a function.Suppose that f is injective.Prove that x~f(x).

Answer 1

Well I can't fully prove this for you, because
1) I suck at proofs and
2) I'm not familiar with the x ~ f(x) notation.

Start with what you know. You are given that f is injective. So you know that if $$f(x_1) = f(x_2) \ \text{then} \ x_1 = x_2, \ \ \ \ \forall x_1, x_2 \in A$$

Best of luck.

Answer 2

~ means x and f(X) have same cardinality

Answer 3

Eh I'm sorry, I've dealt very little with cardinality. I have notes on this in one of my notebooks, but that's at my other house. My apologies!

Answer 4

I would try approaching this by doing a proof by contradiction. Use the definition of what it means to be a function and what it means to be injective. I think you could squeeze something out.

Answer 5

thank you.but it is still vague and very hard to me.hope ı can prove it.

Answer 6

Yeah I realize I wasn't much of a help :/
But I think you can do it! Remember that given an x, a function (by definition) cannot map to two separate f(x) values. So if the function can be evaluated at all x and each f(x) is unique, then x~f(x).

Hopefully that's a little more helpful.

Answer 7

well we'll see...

Answer 8

The standard way to prove that two sets have the same cardinality is to give a bijective function between them.

If I'm understanding your notation correctly, in x~f(x) x refers to the domain of f and f(x) the image.

So this boils down to showing that an injective function is bijective between its domain and its image. Injectivity is in the premise and surjectivity comes from the definition of the image.