hello.
I cant seem to prove the following:
are two different sets such that
prove that if there exists an injective function .
then there exists a countable infinite set
(C and A can overlap)
can someone give me some hints?
thanks.
hello.
I cant seem to prove the following:
are two different sets such that
prove that if there exists an injective function .
then there exists a countable infinite set
(C and A can overlap)
can someone give me some hints?
thanks.
There's something wrong here. First of all there is nothing in your statement to force A to be at least countably infinite, which is a requirement for C to be countably infinite.
For a counter-example let A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6, 7}. Define an injection . Then and f is an injection, but there exists no set that is countably infinite.
-Dan
If is infinite this implies that is also infinite and at least as large as because of the injective map. Now, the property of the integers say there are contained up to cardinality in any infinite set, thus, there exist an injection
Then the image of the function,
(I hope that is what you mean by overlap).
I assume that you are told is an infinite set.
1) is an infinite set also because the injection from implies that in cardinality. So it must be infinite becuase it is larger than another infinite set (it cannot be finite because any infinite set is larger than any finite set).
2)Therefore, there exists an injection map . Because is the smallest possible infinite set which implies its size is contained in any infinite set. Thus, we state that in terms of an injection (since injection shows that one set is less than another in cardinality).
3)The image of the injection (image of a function, you should know what that is) is an infinite set that is the size of the integers and is contained in . Thus, which proves what you were asking.
I can prove that! If that is what he wants.
I shall show there cannot exist a surjection,
where is finite and is infinite.
Assume there is one.
Then since is infinite (definition of infinity).
Then, the inverse image,
is a proper subset of (because that set was proper in the infinite set, the inverse image preserve this). But since we have . Thus, there exists a proper subset of a finite set having the same cardinality which is impossible by the definition of what finiteness is.
This is another case where the uniformity of mathematical definitions fails.
In the above, the definition of ‘infinite set’ was used.
Well which definition. This makes hard to help with knowing the text being followed.
There are at least three or maybe four popular definitions:
A set is infinite if it is not finite. A finite set is equipotent to a natural number.
A set is infinite if it is equipotent to a proper subset of itself (called Dedekind infinite).
A set is infinite if it contains a copy of the natural numbers.
I suspect that the purpose of the question was to show that “If an infinite set is mapped injectively of another set then the final set must also be infinite.” From the wording it is difficult to know what definition is being used. In any case my first response works.