Results 1 to 8 of 8

Thread: Problem 1, Section 1...:(

  1. #1
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,849

    Problem 1, Section 1...:(

    Explain why $\displaystyle 2\in\{1,2,3\}$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1
    Quote Originally Posted by VonNemo19 View Post
    Explain why $\displaystyle 2\in\{1,2,3\}$.
    Is this a trick question?
    What don't understand about it? Do you know the symbols involved?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,849
    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 $\displaystyle 2\in\{1,2,3\}$ is a true statement.

    Right?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790
    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".
    Follow Math Help Forum on Facebook and Google+

  5. #5
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,849
    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?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790
    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.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Banned
    Joined
    Mar 2009
    Posts
    256
    Thanks
    1
    Quote Originally Posted by VonNemo19 View Post
    Explain why $\displaystyle 2\in\{1,2,3\}$.
    Here is the proof:

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

    Since $\displaystyle x\in\{a,b,c\}\Longleftrightarrow x= a\vee x= b\vee x= c$
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Nov 2009
    Posts
    10
    Hello at all!

    Here is a formal proof for the statement:

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

    Best wishes,
    Seppel
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. mean value theorem section problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 11th 2011, 01:21 PM
  2. [SOLVED] Problem from a section on Bezout's Identity
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 18th 2010, 06:27 PM
  3. Replies: 1
    Last Post: Mar 25th 2009, 12:18 PM
  4. Replies: 2
    Last Post: Mar 30th 2008, 07:40 AM

Search Tags


/mathhelpforum @mathhelpforum