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Math Help - Sequences and Summations Proof (Pre-imaging)

  1. #1
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    Sequences and Summations Proof (Pre-imaging)

    Hey, could anyone give me some assistance on this proof?

    Let f be a function from A to B. Let S and T be subsets of B. Show that

    f-1(S U T) = f-1(S)U f-1(T).


    Where the f to the negative one denote the pre-image or inverse image.

    I think by comparing the inverse images of two subsets in a function we might be comparing if they are equivalent, but not sure.

    This comes from the Basic Structures: Sets, Functions, Sequences, and Sums from Sequences and Summations chapter.

    Thanks!
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  2. #2
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    Quote Originally Posted by Saint22 View Post
    Hey, could anyone give me some assistance on this proof?

    Let f be a function from A to B. Let S and T be subsets of B. Show that

    f-1(S U T) = f-1(S)U f-1(T).


    Where the f to the negative one denote the pre-image or inverse image.

    I think by comparing the inverse images of two subsets in a function we might be comparing if they are equivalent, but not sure.

    This comes from the Basic Structures: Sets, Functions, Sequences, and Sums from Sequences and Summations chapter.

    Thanks!

    To prove that f^-1(SUT) = f^-1(S)U f^-1(T) WE must show that :

    .........xε{ f^-1(SUT)} <=====>xε{ f^-1(S)U f^-1(T)}.


    So :

    xε{ f^-1(SUT)}<======> xεA & ( f(x)ε(SUT))<======>xεA& (f(x)εS v f(x)εT) <=====> (xεA & f(x)εS) v ( xεA & f(x)εT)

    <=====> xε f^-1(S) v xε f^-1(T) <=====>xε{ f^-1(S)U f^-1(T)}

    Hence :



    ............................ f^-1(SUT) = f^-1(S)U f^-1(T) .

    This a proof using the double implication procedure,where each step of the forward proof can be reversed;

    Also note that by definition  f^-1(S) = { xεA: f(x)εS } e.t.c,e.t.c

    and if we put ........xεΑ=p..............f(x)εS=q............... ..f(x)εT = r,then


    .........xεA& (f(x)εS v f(x)εT) becomes p&( q v r) which by the distributive property of propositional calculus is equal (p&q) v ( p & r) =


    (xεA & f(x)εS) v ( xεA & f(x)εT)...e.t.c,e.t.c
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  3. #3
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    Hey thanks for the response, but just wondering if "ε" is meant to be denoted as "an element of" or a "subset."

    Thanks again!
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  4. #4
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    Quote Originally Posted by Saint22 View Post
    Hey thanks for the response, but just wondering if "ε" is meant to be denoted as "an element of" or a "subset."

    Thanks again!

    ε ..........is the 1st letter of the Greek word ,i think, ειναι which means belongs to i.e "is an element of"
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