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Let $F$ be the collection of all functions $f: \{1, 2, 3\} \to \{1, 2, 3\}$. If $f$ and $g \in F$, define an equivalence relation $\sim$ by $f\sim g$ if and only if $f(3) = g(3)$. 

  1. Find the number of equivalence classes defined by $\sim$. 
  2. Find the number of elements in each equivalence class. 

5 Answers

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Two functions f and g  in F are said to be Equivalent if and only if f(3)=g(3).

For any function f in F, f(3) can be either 1 or 2 or 3.

When f(3) = 1, Number of such functions possible in F for which f(3) = 1, will be  3^2 = 9 and all these 9 functions for which f(3) = 1, Are Equivalent (according to the condition defined for Equivalence of functions) 

Similarly for, When f(3) = 2  and When f(3) = 3. 

So, there will be Three Equivalence Classes, One for when f(3) = 1, One for when f(3) = 2, and One for f(3) = 3.  And Each equivalence class will have 9 Elements(functions) each.

So, Answer for 

1. : number of equivalence classes defined by "∼"  = 3

2.:  number of elements in each equivalence class  = 9

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