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Exercise 24 - Chapter 3

edited June 2018 in Exercises

Let \( \underline{2} = \{1, 2\} \) and \( \underline{3} = \{1, 2, 3 \} \). These are objects in the category Set. Write down all the elements of the set \( \textbf{Set}( \underline{2}, \underline{3} ) \); there should be nine.

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The category of sets, denoted Set, is defined as follows.

(i) \( Ob(\textbf{Set}) \) is the collection of all sets.

(ii) If \(S\) and \(T\) are sets, then \( \textbf{Set}(S, T) = \{ f : S \rightarrow T | f \text{ is a function } \} \).

(iii) For each set \(S\), the identity morphism is the function \( id_S : S \rightarrow S \) given by \( id_S (s) = s \) for each \( s \in S \).

(iv) Given \( f : S \rightarrow T \) and \( g : T \rightarrow U \), their composite \(f . g\) sends \( s \in S \) to \( g( f (s)) \in U \).

Comments

  • 1.

    The \( | \textbf{Set} (n, m) | = | m |^{|n|} \).

    Comment Source:The \\( | \textbf{Set} (n, m) | = | m |^{|n|} \\).
  • 2.

    \(f_1(1)=f_1(2)=1,f_2(1)=f_2(2)=2,f_3(1)=f_3(2)=3\)

    \(f_4(1)=1,f_4(2)=2,f_5(1)=2,f_5(2)=1,f_6(1)=2,f_6(2)=3\)

    \(f_7(1)=3,f_7(2)=2,f_8(1)=1,f_8(2)=3,f_9(1)=3,f_9(2)=1\)

    Comment Source:\\(f_1(1)=f_1(2)=1,f_2(1)=f_2(2)=2,f_3(1)=f_3(2)=3\\) \\(f_4(1)=1,f_4(2)=2,f_5(1)=2,f_5(2)=1,f_6(1)=2,f_6(2)=3\\) \\(f_7(1)=3,f_7(2)=2,f_8(1)=1,f_8(2)=3,f_9(1)=3,f_9(2)=1\\)
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