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I have the following question: Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B.

I find it hard to prove it because I can easily find a set A which is:

A={1,2}
B={1,2,3,4}
C={1,2,3}

C is a subset of B. C's cardinality is bigger than A's.

There is an injective function from A to B, of course. Assunption: A,B,C are finite. where am I wrong?

Thank you.

Alan
  • 2,791
  • The question is wrong. Are you sure that's what you're asked? – nomen Dec 07 '13 at 17:13
  • The function should be bijective. – Aiden Dec 07 '13 at 17:15
  • You seem to be misreading the question. What the question says is: There is an injection from $A$ to $B$ iff there is some set of $B$ that is in bijection with $A$. It does not say "iff every subset of $B$ is in bijection with $A$". That is, the fact that there is a subset $C$ of $B$ larger than $A$ does not matter. In your example, yes, $C$ is larger than $A$, but $A$ is a subset of $B$ and clearly $A$ has the same size as $A$. – Andrés E. Caicedo Dec 07 '13 at 17:15
  • @Aiden I agree. That'a what's wierd here. – Alan Dec 07 '13 at 17:15
  • Also, you seem to be assuming that the sets involved are finite, or at least $A$ is finite. If you are making that assumption implicitly (as the title suggests), please add it to the question. – Andrés E. Caicedo Dec 07 '13 at 17:17
  • (You may think of it this way: Say that $A$ has size $n$. Suppose there is an injection from $A$ into $B$. Then we must have that if $|B|=m$, we have $m\ge n$. In your example, $A$ has size $2$, $B$ has size $4$. Any set of size $4$ has a subset of size $2$. It also has subsets of size $3$, but that is irrelevant for this problem.) – Andrés E. Caicedo Dec 07 '13 at 17:19
  • @AndresCaicedo so you're saying that I have to find that a subset of B exists that it has same Cardinality of A? – Alan Dec 07 '13 at 17:21
  • Yes, under the assumption that there is an injection from $A$ to $B$, you need to show that there is a subset of $B$ of the same cardinality as $A$. That is half of the problem. The other half is to show the converse, that is, if there is a subset of $B$ of the same cardinality as $A$, then there is an injection from $A$ into $B$. – Andrés E. Caicedo Dec 07 '13 at 17:22
  • Thank you! @AndresCaicedo. btw, edited as you said. Now I understand ''iff'' better. – Alan Dec 07 '13 at 17:23

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