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Thread: finite cardinality proof

  1. #1
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    Exclamation finite cardinality proof

    Prove that {1,...,10}X{1,...,15} is finite using only the definition of a finite set.
    Defn.: "a set S is finite if either S is the empty set (well obviously not!) OR S is equivalent to the set {1,2,3,...,n} for some positive integer n.
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  2. #2
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    Quote Originally Posted by dolphinlover View Post
    Prove that {1,...,10}X{1,...,15} is finite using only the definition of a finite set.
    Defn.: "a set S is finite if either S is the empty set (well obviously not!) OR S is equivalent to the set {1,2,3,...,n} for some positive integer n.
    Consider $\displaystyle fa,b) \to \left[(a-1)+10\times (b-1) +1\right] $

    Then the smallest value $\displaystyle f$ takes is $\displaystyle 1$, and the largest is $\displaystyle 150$, the image of distinct elements of $\displaystyle \{1,...,10\}X\{1,...,15\}$ are distinct. Thus we have $\displaystyle 150$ points in the range and $\displaystyle 150$ points in the image, and the image is: $\displaystyle \{1, ..., 150\}$.

    Now turn this into a formal demonstration.

    CB
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  3. #3
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    Quote Originally Posted by Mathstud28 View Post
    Well if $\displaystyle f:[1,10]\mapsto[1,15]$ your domain would be $\displaystyle [1,10]$ and your codomain would be $\displaystyle [1,15]$. So now before I say something that is unclear again, do you know what either one-to-one or mapping means? Do you understand what it means if two sets are equivalent, denoted $\displaystyle A\sim{J}$
    Yes I do understand 1-to-1 & onto (mapping)...Most of our proofs in the last couple weeks have involved bijection....how does this relate to whether or not it's finite (or infinite)...I have already defined my domain & codomain (see previous entry)... f: domain--->codomain. What is f?...graphically it's a solid rectangle therefore how is it 1-to-1? (the cartesian product)....I need help!
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