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Consider a composite $f \circ g(x) = f(g(x))$ of two maps $X \xrightarrow{g} Y \xrightarrow{f} Z$. If $f$ is injective and $g$ is surjective, what is the results of the composition $f \circ g$? [or DNE if it does not always have one of these properties]

Similarly, bijective and surjective, AND bijective and injective?

Thank you!

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What's the definition of each? Just trying to see if you know what each mean. –  Amateur Math Guy Feb 23 '13 at 18:54
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why did you tag it with number theory? –  Dominic Michaelis Feb 23 '13 at 18:56
    
Welcome to MSE. Please note that LaTeX is the preferred way to write Maths here. Take a look at the editing I did on your question to see some details. Also, one expects to see some kind of progress you have done on your questions. Have you tried constructing examples? –  Andreas Caranti Feb 23 '13 at 19:00
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1 Answer

A hint on your first question. Try and take $X = Z = \{1, 2\}$ and $Y = \{3\}$.

For your other questions, try some examples, and then decide whether you want to prove something.

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