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I know that $f : A \to B$ means the function is mapping $A$ to $B$.

My question is that when I say cardinality of $B$ is $x$, then does the x mean total numbers of elements present in $B$ or does it mean that total number of elements getting mapped?

Thank You Any help appreciated.

user
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MacVimHelp
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1 Answers1

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For finite sets, the carinality of $B$ represents the number of elements which can be potentially but not necessarly reached by $f$.

When all elements of $B$ are reached we say that $f$ is a surjective function.

user
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  • so if the cardinality of B is same as A and the function is surjective then can I say that the function is injective as each element is getting mapped and is unique, therefore it is bijective? (Does this remains same if the A and B is set of natural numbers and both are same) – MacVimHelp Oct 26 '18 at 22:04
  • @MacVimHelp Yes for finite sets it is absolutely correct! Not for $\mathbb{N}$. – user Oct 26 '18 at 22:10
  • @MacVimHelp Think to $f(1)=1, f(2)=1, f(3)=2,...,f(n)=n-1$. – user Oct 26 '18 at 22:11
  • SN is a set {1, 2, 3, . . . , N }, if a map f : SN -> SN is surjective, then f is a bijection. Does this mean that this statement is false? – MacVimHelp Oct 26 '18 at 22:14
  • If I am not wrong then this has N numbers so the statement is true, amirite? – MacVimHelp Oct 26 '18 at 22:16
  • @MacVimHelp When A and B are finite sets and $|A|=|B|$ the statement is (always) true, but if $A=B=\mathbb{N}$ it is not true (in general). – user Oct 26 '18 at 22:18
  • [link] https://math.stackexchange.com/questions/2971410/argue-that-if-a-map-on-a-finite-set-is-surjective-then-it-is-a-bijection/2971510#2971510 – MacVimHelp Oct 26 '18 at 22:19
  • @MacVimHelp What exactly it is not clear to you. – user Oct 26 '18 at 22:25
  • @MacVimHelp I've already explained it in detail. Are you interested in the case with $|A|=|B|$ with $A$ and $B$ finite? – user Oct 26 '18 at 22:33
  • lol Sorry for confusing you, I am working on this statement SN is a set {1, 2, 3, . . . , N }, if a map f : SN -> SN is surjective, then f is a bijection. – MacVimHelp Oct 26 '18 at 22:38
  • @MacVimHelp Yes of course that's true, are you looking for a prove about that fact? – user Oct 26 '18 at 22:39
  • Never mind I was mixing up ℕ and N, thanks a lot for helping me out – MacVimHelp Oct 26 '18 at 22:43
  • @MacVimHelp Ah ok! You are welcome! Bye – user Oct 26 '18 at 22:44