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Given the next sets: $A=\{1,2\}$ , $B=\mathbb{N}$,

a) Describe all the functions from $A$ to $B$ that are not injective.

b) Describe all the functions from $B$ to $A$ that are not onto.

c) Prove or disprove the next:

(i) The next functions exist $f\colon A\to B$ and $g\colon B\to A$ if $g\circ f$ reversal

(ii) the next functions exist $f\colon A\to B$ and $g\colon B\to A$ if $f\circ g$ reversal

Need help here. Thanks.

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    Where are you stuck?2010-12-29
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    What does «if $g\circ f$ reversal» mean?2010-12-29
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    This seems kind of homework-y to me, but I'll give hints. For (a), what does it mean for something to be injective? We have that f(a) = f(b) implies a = b. Since we only have two elements here, if we send one of them to an element in $N$, then where does the second one have to go? For (b) ON, I guess, means surjective. Surjectivity means that the function "hits" all the elements in A. Where would we have to send elements in B to in order for them to all "miss" one element in A?2010-12-29
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    reversal in like reverse the function the other way i think and im stuck everywhere need some answers here2010-12-29
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    For (c), I think you are trying to ask if there exist functions $f$ and $g$ so that (i) $g \circ f$ is bijective (invertible) or (ii) $f \circ g$ is bijective. Remember, a function is bijective if and only if it is both injective and surjective.2010-12-29

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