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Thread: What does this even mean? So confused...

  1. #1
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    What does this even mean? So confused...

    I am totally stuck on this problem.

    Suppose that $\displaystyle A$ is a subset of $\displaystyle V^{*}$, where $\displaystyle V$ is an alphabet. Prove or disprove each of these statements.

    a) $\displaystyle A \subseteq A^2$
    b) $\displaystyle \mbox{If} \ A = A^2, \mbox{then} \ \lambda \in A$
    c) $\displaystyle A\{\lambda\} = A$
    d) $\displaystyle (A^{*})^{*} = A^{*}$
    What do all these symbols even mean i.e. *, lambda, etc. and how do I go about in solving these problems? Thank you.
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  2. #2
    Newbie aleph1's Avatar
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    Quote Originally Posted by VENI View Post
    I am totally stuck on this problem.

    Suppose that $\displaystyle A$ is a subset of $\displaystyle V^{*}$, where $\displaystyle V$ is an alphabet. Prove or disprove each of these statements.

    a) $\displaystyle A \subseteq A^2$
    b) $\displaystyle \mbox{If} \ A = A^2, \mbox{then} \ \lambda \in A$
    c) $\displaystyle A\{\lambda\} = A$
    d) $\displaystyle (A^{*})^{*} = A^{*}$
    What do all these symbols even mean i.e. *, lambda, etc. and how do I go about in solving these problems? Thank you.
    I am not sure. Doing a wiki search on "alphabet subset theory" I came up with this.

    Free monoid - Wikipedia, the free encyclopedia

    $\displaystyle V^{*}$ may mean an alphabet set minus the empty string.

    $\displaystyle \lambda$ may be the empty string.

    Guessing, $\displaystyle A^2$ may be the 2-tuples of elements of A.

    With the above, I further conjucture weakly:

    a) is false. $\displaystyle A \$ is not a 2-tuple.
    b) is false. $\displaystyle A \$ is not a 2-tuple. $\displaystyle A = A^2 \$ does not result in $\displaystyle \ \ \lambda \in A$
    c) not sure what $\displaystyle A\{\lambda\}$ implies.
    d) is true. Removing the empty string twice from the set $\displaystyle A$ which already does not contain the empty string, is equivalent to removing the empty string once.

    Supply more context on the source of this, for example: title of book, paragraph, section or chapter, or class topic under discussion.
    Last edited by aleph1; Apr 23rd 2009 at 11:12 AM.
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  3. #3
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    Quote Originally Posted by aleph1 View Post
    I am not sure. Doing a wiki search on "alphabet subset theory" I came up with this.

    Free monoid - Wikipedia, the free encyclopedia

    $\displaystyle V^{*}$ may mean an alphabet set minus the empty string.

    $\displaystyle \lambda$ may be the empty string.

    Guessing, $\displaystyle A^2$ may be the 2-tuples of elements of A.

    With the above, I further conjucture weakly:

    a) is false. $\displaystyle A \$ is not a 2-tuple.
    b) is false. $\displaystyle A \$ is not a 2-tuple. $\displaystyle A = A^2 \$ does not result in $\displaystyle \ \ \lambda \in A$
    c) not sure what $\displaystyle A\{\lambda\}$ implies.
    d) is true. Removing the empty string twice from the set $\displaystyle A$ which already does not contain the empty string, is equivalent to removing the empty string once.

    Supply more context on the source of this, for example: title of book, paragraph, section or chapter, or class topic under discussion.
    Thank you for your help. The problem comes from a hand out my professor gave me about finite-state automata.
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