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Math Help - Power Sets

  1. #1
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    Power Sets

    Prove: Let S and T be sets. If there exists and injection from |S| --> |T| then there exists an injection from |P(S)|-->|P(T)|


    (P(S) and P(T) represent the power sets of each)
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  2. #2
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    Here is a bit of notation. C \subseteq S \Rightarrow \quad \overrightarrow f (C) = \left\{ {t \in T:\left( {\exists x \in C} \right)\left[ {f(x) = t} \right]} \right\}
    That is called the image set of C under f.
    Now it is clear that we have \overrightarrow f :P(S) \to P(T), so we must prove that if f\mbox { is injective then } \overrightarrow f \mbox{ is also injective}
    Of course to do that you must show \overrightarrow f (C) = \overrightarrow f (D) \Rightarrow \quad C = D.
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  3. #3
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    Power Sets

    The proof is regarding the cardinality of the sets and power sets. Does that change the strategy?
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  4. #4
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    Quote Originally Posted by GoldendoodleMom View Post
    The proof is regarding the cardinality of the sets and power sets. Does that change the strategy?
    No it does not.
    Here is the rest of it.
    If \overrightarrow f (C) = \overrightarrow f (D) then
    \begin{array}{lcl}<br />
   {c \in C} &  \Rightarrow  & {f(c) \in \overrightarrow f (C)}  \\<br />
   {} &  \Rightarrow  & {f(c) \in \overrightarrow f (D)}  \\<br />
   {} &  \Rightarrow  & {\left( {\exists d \in D} \right)\left[ {f(d) = f(c)} \right]}  \\<br />
   {} &  \Rightarrow  & {d = c}  \\<br />
   {} &  \Rightarrow  & {c \in D}  \\<br /> <br />
 \end{array}
    You do the other way by switching D with C.
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