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Math Help - Problem 1, Section 1...:(

  1. #1
    No one in Particular VonNemo19's Avatar
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    Problem 1, Section 1...:(

    Explain why 2\in\{1,2,3\}.
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  2. #2
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    Quote Originally Posted by VonNemo19 View Post
    Explain why 2\in\{1,2,3\}.
    Is this a trick question?
    What don't understand about it? Do you know the symbols involved?
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Plato View Post
    Is this a trick question?
    What don't understand about it? Do you know the symbols involved?
    I got it. I had to read the section again.

    In particular:

    "According to cantor, a set is made up of objects called members or elements...If the first blank in '_____ is a member of _____' is filled in with the name of an object, and the second with the name of a set, the resulting sentence can be classified as either true or false."

    So, I guess the answer is

    Because 2\in\{1,2,3\} is a true statement.

    Right?
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  4. #4
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    Explain why
    Because is a true statement.
    The questions "Why ?" and "Why '' is a true statement" are, in fact, one and the same. If somebody asks you, "Why are you late?", it does not make sense to answer "Because 'I am late' is a true statement".
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  5. #5
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by emakarov View Post
    The questions "Why ?" and "Why '' is a true statement" are, in fact, one and the same. If somebody asks you, "Why are you late?", it does not make sense to answer "Because 'I am late' is a true statement".
    So, what would you say they are looking for here?
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  6. #6
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    I am also not sure about exact words for such a trivial case, but I would say, because 2 is listed as an element of the set. That's why it belongs.

    Unless it's advanced math, the concepts of a set, an element and belonging are taken as primitive.
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  7. #7
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    Quote Originally Posted by VonNemo19 View Post
    Explain why 2\in\{1,2,3\}.
    Here is the proof:

    2=2\Longrightarrow 2=2\vee 2=1\vee 2=3\Longrightarrow 2=1\vee\ 2=2\vee 2=3\Longrightarrow 2\in\{1,2,3\}.

    Since  x\in\{a,b,c\}\Longleftrightarrow x= a\vee x= b\vee x= c
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  8. #8
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    Hello at all!

    Here is a formal proof for the statement:

    1) \forall x(x=x).................................................. .................................................. .law of identity
    2) 2=2.................................................. .................................................. ............1, Universal Elimination, where we put x=2
    3) 2=2\vee 2=1.................................................. .................................................. ....2, Disjunction Introduction
    4) 2=1\vee 2=2.................................................. .................................................. ....3, Commutation equivalence
    5) \forall x\forall y\forall z[z\in \{x,y\}\Longleftrightarrow z=x\vee z=y].................................................. ...........theorem of set theory
    6) 2\in \{1,2\}\Longleftrightarrow 2=1\vee 2=2.................................................. ......................5, Universal Elimination, where we put x=1, y=2 and z=2
    7) 2=1\vee 2=2\Longrightarrow 2\in \{1,2\}.................................................. ..........................6, Biconditional Elimination
    8) 2\in \{1,2\}.................................................. .................................................. ...4, 7, Modus Ponens
    9) 2\in \{1,2\}\vee 2\in \{3\}.................................................. ......................................8, Disjunction Introduction
    10) \forall x\forall A\forall B[x\in A\cup B\Longleftrightarrow x\in A\vee x\in B].................................................. ....theorem of set theory
    11) 2\in \{1,2\}\cup \{3\}\Longleftrightarrow 2\in \{1,2\}\vee 2\in \{3\}.............................................10, Universal Elimination, where we put x=2, A=\{1,2\} and B=\{3\}
    12) 2\in \{1,2\}\vee 2\in \{3\}\Longrightarrow 2\in \{1,2\}\cup \{3\}.................................................1 1, Biconditional Elimination
    13) 2\in \{1,2\}\cup \{3\}.................................................. ..........................................9, 12, Modus Ponens
    14) \forall x\forall y\forall z[\{x,y,z\}=\{x,y\}\cup \{z\}].................................................. ........definition of triplet set
    15) \{1,2,3\}=\{1,2\}\cup \{3\}.................................................. .....................................14, Universal Elimination, where we put x=1, y=2 and z=3
    16) \forall X\forall Y[X=Y\Longleftrightarrow \forall x(x\in X\Longleftrightarrow x\in Y)]............................axiom of extensionality
    17) \{1,2,3\}=\{1,2\}\cup \{3\}\Longleftrightarrow \forall x[x\in \{1,2,3\}\Longleftrightarrow x\in \{1,2\}\cup \{3\}]..............16, Universal Elimination, where we put X=\{1,2,3\} and Y=\{1,2\}\cup \{3\}
    18) \{1,2,3\}=\{1,2\}\cup \{3\}\Longrightarrow \forall x[x\in \{1,2,3\}\Longleftrightarrow x\in \{1,2\}\cup \{3\}]..............17, Biconditional Elimination
    19) \forall x[x\in \{1,2,3\}\Longleftrightarrow x\in \{1,2\}\cup \{3\}]................................................15 , 18, Modus Ponens
    20) 2\in \{1,2,3\}\Longleftrightarrow 2\in \{1,2\}\cup \{3\}.................................................. ........19, Universal Elimination, where we put x=2
    21) 2\in \{1,2\}\cup \{3\}\Longrightarrow 2\in \{1,2,3\}.................................................. ............20, Biconditional Elimination
    22) 2\in \{1,2,3\}.................................................. .................................................. 13, 21, Modus Ponens

    Best wishes,
    Seppel
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