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Math Help - Set Theory: Functions (injectivity/surjectivity)

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    Question Set Theory: Functions (injectivity/surjectivity)

    Suppose f : A -> B and g : B -> C are functions.
    (a) Show that if f is surjective and g is not injective then g o f is not injective.
    (Hint: draw pictures to get an idea.)
    (b) Show that if f is not surjective and g is injective then g o f is not surjective.


    hey, I have a general idea how to solve questions like this, but these ones seem more challenging than any ive done so far, so any help is is really appreciated
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  2. #2
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    Quote Originally Posted by zukias View Post
    Suppose f : A -> B and g : B -> C are functions.
    (a) Show that if f is surjective and g is not injective then g o f is not injective.
    (Hint: draw pictures to get an idea.)
    (b) Show that if f is not surjective and g is injective then g o f is not surjective.
    From the given we can get the following:
    \left( {\exists b_1  \in B} \right)\left( {\exists b_2  \in B} \right)\left[ {b_1  \ne b_2  \wedge g(b_1 ) = g(b_2 )} \right].

    ALSO \left( {\exists a_1  \in A} \right)\left[ {f(a_1 ) = b_1 } \right] \wedge \left( {\exists a_2  \in A} \right)\left[ {f(a_2 ) = b_2 } \right].

    There is an easy contradiction to injectivity there for g\circ f.
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