Options

# Exercise 24 - Chapter 3

edited June 2018

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.

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 $$g( f (s)) \in U$$.

• Options
1.

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

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