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Math Help - example of a function (one to one & onto)

  1. #1
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    example of a function (one to one & onto)

    Give an example of a function from N to N that is one to one but not onto

    My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 3, f(b) = 4, f(c) = 1. Is this the correct example to this question? What does it mean from N to N?
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    Quote Originally Posted by TheRekz View Post
    Give an example of a function from N to N that is one to one but not onto

    My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 3, f(b) = 4, f(c) = 1. Is this the correct example to this question? What does it mean from N to N?
    f(1)=2, f(2)=3, f(3)=4, ... f(n) = n+1, ...
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    Quote Originally Posted by TheRekz View Post
    Give an example of a function from N to N that is one to one but not onto
    My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 3, f(b) = 4, f(c) = 1. Is this the correct example to this question? What does it mean from N to N?
    No it is not correct.
    First N is the set of counting numbers either {0,1,2,3,...} or {1,2,3,4,...} (i.e. it contains 0 or not depending on your textbook). So N is infinite; so your example must be infinite also. Here is another example in addition to the one given above.
    f:N \mapsto N,\quad \left( {n \in N} \right)\left[ {f(n) = 2^n } \right]\quad
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    Note if S is a finite set it is NOT POSSIBLE to find:

    f:S\to S that is one-to-one but not onto. This is a consequence of the Pigeonhole Principle. This is a distinction between finite and infinite sets.
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    Quote Originally Posted by Plato View Post
    No it is not correct.
    First N is the set of counting numbers either {0,1,2,3,...} or {1,2,3,4,...} (i.e. it contains 0 or not depending on your textbook). So N is infinite; so your example must be infinite also. Here is another example in addition to the one given above.
    f:N \mapsto N,\quad \left( {n \in N} \right)\left[ {f(n) = 2^n } \right]\quad
    so what you're saying is that the function f(x) = 2^x will work as an example of this question?
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    Quote Originally Posted by TheRekz View Post
    so what you're saying is that the function f(x) = 2^x will work as an example of this question?
    Yes. The function I gave it even simpler.
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    What if the question is neither one to one nor onto? What would be a perfect example? Would x^2 + 1 work?
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    Quote Originally Posted by TheRekz View Post
    What if the question is neither one to one nor onto? What would be a perfect example? Would x^2 + 1 work?
    x^2+1 is one-to-one in this case.

    How about,
    f(1)=1
    f(2)=1
    f(3)=1
    f(4)=1
    ...
    f(n)=1
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    Quote Originally Posted by TheRekz View Post
    What if the question is neither one to one nor onto? What would be a perfect example? Would x^2 + 1 work?
    No! Because the elements of N are non-negative that function is one-to-one.
    Look at the function f:N \mapsto N,\quad f(n) = \text{floor}\left( {\frac{{n + 5}}{2}} \right)
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    can you give me other examples that does not use the floor operator?
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    N is a set of natural numbers right? And a natural number can't be negative?
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    Quote Originally Posted by TheRekz View Post
    N is a set of natural numbers right? And a natural number can't be negative?
    correct. the natural numbers is the set of positive integers: 1,2,3,4,5,6....


    wow, i could actually answer a question in this thread
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    Quote Originally Posted by TheRekz View Post
    N is a set of natural numbers right? And a natural number can't be negative?
    Look! We do not know what textbook or set of notes you are following.
    Sorry to say, but it is true nonetheless, there are no hard and fast definition in mathematics. So you read the definition of N in your text material. Most texts that I have seen require N to be either {0,1,2,3,…} or {1,2,3,…}*. So yes the customary and usual definitions of N mean that the set is made of non-negative integers.

    * See this site: Counting Number -- from Wolfram MathWorld
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