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Math Help - Need help on probablity class!!!

  1. #1
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    Post Need help on probablity class!!!

    Hi, I'm taking probability class. We only use materials that is provided by our instructure and we dont use book. It's kind of hard for me to understand what is the concept because there is no example is given that I can related to to concept.
    I'm having trouble with
    1. Provide, if possible, an example of an object collection that is infinite.
    2. What is meant by the statement that an object collection M is not infinite?

    Now, before helping me, here is what is given.
    I) The statement that P is an ordered object pair means there is an object a, called the first object, and there is an object b, called the second object. We shall denote such an ordered object pair by (a,b). We may call a the first element of the ordered pair and b the second of the ordered pair.
    So far so good. Completely making sense to me.
    II) Suppose each of A and B is a set. The statement that f is a function from A into B means:
    a) f is an ordered element pair collection;
    b) only (a,b) belongs to f when a is in A and b is in B;
    c) No two such ordered pairs have the same first element.
    This is where it starts to not make sense to me...there is no example that shows why that is true and I can't exactly prove it even though this is a rule.
    Let's continue.
    The set to which only x belongs when x is the first element of an ordered pair belonging to f is called the initial set, or domain, of f and is denoted by D sub f.
    The set to which only y belongs when y is the second element of an ordered pair in f is called the final set, or range, of f and is denoted by R sub f.
    If (u,v) is an ordered pair in f then we may denote v by f sub u or f(u) and the pair (u,v) by (u, f sub u) or (u, f(u)).
    If D sub f is a collection of one or more sets, that is, if each member of D sub f is a set, then f is said to be a set function.
    Finally, if every element of set B is in R sub f, then f is said to be from A onto B.
    The statement that an object set S is infinite means that if n is any positive integer greater than 1, then S has (at least) n objects.

    At this point, I can't even make sense of any of this without understanding what the 3 points mean earlier in this message. Can anyone help me?
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  2. #2
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    Re: Need help on probablity class!!!

    Hey dburneebaatar.

    With regard to your question, an example of a finite set is {(1,2),(1,3),(1,4)} and an infinite set is X = {x e X: x is a Real number in [0,1]}
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  3. #3
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    Re: Need help on probablity class!!!

    I wouldn't be too happy with the instructor. It appears to me as though he is trying to formalize some very basic notions such as what is meant by a function f from a set A to a set B. First, in my book, the stated definition of ordered pair is not precise -- I think usually the definition of (a,b) is {{a},{a,b}}, which is precise but almost incomprehensible.

    Formally, a function f from a set A to a set B is a subset of A\times B (the set of all ordered pairs (a,b) with a in A and b in B). The subset f must satisfy:
    1. For every a in A, there is a b in B with (a,b) in f.
    2. For any a in A, b and b' in B with both (a,b) in f and (a,b') in f, it must be that b = b'; i.e. for any a in A, there is exactly one ordered pair (a,b) in f.

    If f is a function from A to B, for any a in A, write f(a) for that unique element of B with (a,f(a)) in f.

    Examples: Let A={1,2,3} and B={4,5}
    1. f={(1,4), (2,5)} is not a function from A to B since no pair has first component 3.
    2. f={(1,4), (2,5), (3,4)} is a good function from A to B.
    3. f={(1,4),(2,4),(2,5),(3,4)} is not a function from A to B since (2,4) and (2,5) are in f, violating 2 of the definition.

    What is meant by a set has "at least n elements"? If you're going to be formal, you need a definition of this. Even more basic what do you mean by sets A and B have the same number of elements? Here, you need the notions of one to one function and onto function. A function f from A to B is one to one provided for any a, a' in A, if f(a)=f(a'), then a=a'. The function f is onto B provided for any b in B, there is an a in A with f(a)=b. So now sets A and B have the same cardinality (same number of elements) if and only if there is a function from A to B which is one to one and onto B.

    I think the usual definition of infinite set is: Let N be the set of natural numbers (integers greater than or equal to 0). The set B is infinite iff there is a one to one function f from N to B (I'm not saying f is necessarily onto B; i.e. it need not be that N and B have the same cardinality.)

    With considerable effort and defining what is meant by the set B has at least n elements, one can prove the equivalence of the above definition with your supposed formal definition.

    One usually then defines a set B to be finite iff B is not infinite. For any natural number k, let Nk = {natural numbers n with n less than or equal to k}. For example N5={0,1,2,3,4,5}. Again with considerable effort, one can prove that a set B is finite iff B is empty or there is some natural number k such that Nk and B have the same cardinality.

    Phew. After all that, here's answers to your specific questions:
    1. The set of natural numbers N is infinite -- define the function f from N to N by: for any n in N, f(n)=n. It's easy to prove f is one to one.

    2. By definition, a set that is not infinite is finite.
    Last edited by johng; February 15th 2013 at 11:15 PM.
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