By Maud King Murphy

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23 1. Set theory In other words, the equivalence set xj R of the quotient set X /R consists of all those arguments in X which have the same value by the function cp. Definition. 5). 5 Conversely, associated with each quotient set there is a surjection. If a quotient set Q of a set X be given we can define a function

F E xx is invertible if and only if f is a bijection. PROOF. 6. D We describe the property Ix of = f o Ix = f for all f E xx by saying I x is a neutral element for composition of functions. The result on invertibility of functions shows us that if we desire to have all functions have in verses we must limit ourselves to bijections. We now do just that. We use the symbol 6(X), an upper case German ess, for the set of all bijections of the set X. Definition. 6(X) = Ulf E xx andf is a bijection}. Because the composition of two bijections is a bijection, I x is a bijection, and the inverse of a bijection is a bijection, we have this theorem: Theorem.

5 Conversely, associated with each quotient set there is a surjection. If a quotient set Q of a set X be given we can define a function

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A Beginner's Guide to the Stars by Maud King Murphy


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