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Math Help - Definition of a sequence of elements in a set?

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    Definition of a sequence of elements in a set?

    Let A denote a non-empty set. Explain what we mean by a sequence of elements in set A.

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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by MathStudent1 View Post
    Let A denote a non-empty set. Explain what we mean by a sequence of elements in set A.

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    By "a sequence of elements in set A" we mean an ordered list of all the elements in A. That is, if we call the first element a_1, the second a_2, the third a_3 and so on, then the sequence of elements in set A is the sequence

    {An}= a_1, a_2, a_3, a_4,....

    for more on sequences see Sequence
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    Quote Originally Posted by MathStudent1 View Post
    Let A denote a non-empty set. Explain what we mean by a sequence of elements in set A.

    Thanks!
    A list of terms belonging to this set.

    Say A={1,A,#) then below is an example of a sequence,

    1,A,A,A,#,#,1,....

    But let us give a formal definition below.
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    is up to his old tricks again! Jhevon's Avatar
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    Note that we can think of a sequence as a function with domain the natural numbers and range some subset of the real numbers. thinking about it that way, if we can find some function of n, where n represents the elements of the natural numbers, then An (that is A sub n), will be the formula for this sequence. plugging in n=1 into the function we get a_1, plugging in n=2 into the function we get a_2 and so on.
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    Quote Originally Posted by Jhevon View Post
    Note that we can think of a sequence as a function with domain the natural numbers and range some subset of the real numbers. thinking about it that way, if we can find some function of n, where n represents the elements of the natural numbers, then An (that is A sub n), will be the formula for this sequence. plugging in n=1 into the function we get a_1, plugging in n=2 into the function we get a_2 and so on.
    I need to warn you. Our Analysis book, accepts that set N (natural numbers) are {1,2,3...}.

    In fact, the standard notion of natural number includes zero as well.

    This is why I used Z^+.

    ---

    And when you said "subset" you should have said "non-trivial subset". Again our book does never says that.
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    Quote Originally Posted by ThePerfectHacker View Post
    I need to warn you. Our Analysis book, accepts that set N (natural numbers) are {1,2,3...}.

    In fact, the standard notion of natural number includes zero as well.

    This is why I used Z^+.

    ---

    And when you said "subset" you should have said "non-trivial subset". Again our book does never says that.
    yeah, text books neglect to mention a lot of stuff.

    i actually had no idea 0 was a natural number
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