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Thread: Advanced Calculus Help

  1. #1
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    Advanced Calculus Help

    A) For each part, find a function f: R -> R that has the desired properties: neither onto nor one-to-one

    B)Under what conditions does A\(A\B) = B?

    C)Define f:J -> N(natural numbers) where f(n) = 2n-1 for each n element of N(natural numbers).

    D) Given A = {1,2,3,4,5}, B = {2,3,4,5,6,7} and C = {a,b,c,d,e} state an example of f: A ->B, g: B-> C, such that g(f)(the composition of g onto f) is 1-1 but g is not 1-1.
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    Talking

    Quote Originally Posted by algebrapro18 View Post
    B)Under what conditions does A\(A\B) = B?
    See #2 of http://www.mathhelpforum.com/math-he...ts-proofs.html by kathrynmath.
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    I read that thread and still can't see the answer.
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    Senior Member bkarpuz's Avatar
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    Question Solution for B

    As we have msntioned previously, $\displaystyle A\backslash B=A\cap B^{c}$, where $\displaystyle B^{c}$ is the compliment set of $\displaystyle B$.
    Noting that $\displaystyle (B^{c})^{c}=B$, and simplifying the given expression, we get
    $\displaystyle A\backslash(A\backslash B)=A\backslash(A\cap B^{c})$
    ..............$\displaystyle =A\cap(A\cap B^{c})^{c}$
    ..............$\displaystyle =A\cap(A^{c}\cup B)$
    ..............$\displaystyle =\underset{\emptyset}{\underbrace{(A\cap A^{c})}}\cup(A\cap B)$
    ..............$\displaystyle =A\cap B.\qquad(*)$
    Since (*) is equal to $\displaystyle B$, we infer that $\displaystyle B\subset A$ is true.
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    Thanks, that helps me with part B now all I need help with is C because I got A and D done by my self.
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    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by algebrapro18 View Post
    Thanks, that helps me with part B now all I need help with is C because I got A and D done by my self.
    Can you please explain C a little bit more?
    What is J or should we find what it is?
    Also do we still want the function f not to be onto and into again?
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    C)Define f:J -> N(natural numbers) where f(n) = 2n-1 for each n which is an element of N.

    I need to use what was given(the line above is all that is given) to find: the image of f, if f is 1-1, if f is onto, and if f is onto I need to find its inverse, the domain of the inverse, and the range of the inverse.

    here is what I have so far:

    The Im(f) has to be all odd numbers because that is what you get when you plug numbers into f(n)=2n-1.

    from there I get stumped.
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    Senior Member bkarpuz's Avatar
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    Lightbulb Solution for C

    Quote Originally Posted by algebrapro18 View Post
    C)Define f:J -> N(natural numbers) where f(n) = 2n-1 for each n element of N(natural numbers).
    Although the following remark is not applicable for this exercise, I would like to tell it.

    Remark. Let $\displaystyle f:A\to B$ be a function and $\displaystyle A,B$ be finite sets if f is onto, then it is one-to-one; vice versa, if f is one-to-one, then it is onto.

    if $\displaystyle J\not\subset K:=\Big\{1,\frac{3}{2},2,\frac{5}{2},\ldots\Big\}$, then $\displaystyle f$ can not be a function with an image which is a subset of $\displaystyle \mathbb{N}$ (pick an element which is not in the set $\displaystyle K$, and see that it is mapped into the set $\displaystyle \mathbb{R}\backslash\mathbb{N}$).
    Therefore, we must have $\displaystyle J\subset K$.
    If $\displaystyle J=K$, then $\displaystyle f$ is one-to-one and onto.
    If $\displaystyle J\neq K$, then $\displaystyle f$ is only one-to-one.

    Just, try to figure it out by yourself by letting $\displaystyle f(n)=2n-1=m$, where $\displaystyle n\in K$ and $\displaystyle m\in\mathbb{N}$. Note that $\displaystyle f$ is strictly increasing, which indicates that it is one-to-one.
    Then obtain the set $\displaystyle K$ by isolating $\displaystyle n$...
    Last edited by bkarpuz; Sep 11th 2008 at 10:41 AM. Reason: Title added.
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  9. #9
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    In english please...I understood about 1 persent of what you said...
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  10. #10
    Senior Member bkarpuz's Avatar
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    Exclamation In English

    Quote Originally Posted by algebrapro18 View Post
    In english please...I understood about 1 persent of what you said...
    If $\displaystyle f$ is a function from $\displaystyle J$ to $\displaystyle \mathbb{N}$ defined by $\displaystyle f(n)=2n-1$, we see that $\displaystyle 2n-1\in\mathbb{N}$ for all $\displaystyle n\in J$.
    This means that for every $\displaystyle n\in J$, there exists $\displaystyle m\in\mathbb{N}$ such that $\displaystyle 2n-1=m$ holds.
    Note that for every $\displaystyle m\in\mathbb{N}$, we may not find $\displaystyle n\in J$.
    Therefore, we see that
    $\displaystyle
    n=\frac{m+1}{2}\text{ for }m\in\mathbb{N}
    $
    holds.
    Since $\displaystyle m\in\mathbb{N}$, we this indicates that
    $\displaystyle
    n\in K:=\Big\{1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2} ,\ldots\Big\}.
    $
    Hence, the domain of $\displaystyle f$ can be picked to be any subset of $\displaystyle K$, i.e. $\displaystyle J\subset K$.
    Now consider the following possible cases.
    If $\displaystyle J=K$, then $\displaystyle f$ is one-to-one and onto.
    If $\displaystyle J\neq K$, then $\displaystyle f$ is only one-to-one.

    Hint. $\displaystyle f$ is strictly icreasing, and it is hence one-to-one.

    I guess it is more clear now?
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