Exercise 1.5.5

Question

\begin{enumerate}[label=(\alph*)] 
  \item Why is $A \sim A$ for every set $A$?
  \item Given sets $A$ and $B$, explain why $A \sim B$ is equivalent to asserting $B \sim A$.
  \item For three sets $A, B$, and $C$, show that $A \sim B$ and $B \sim C$ implies $A \sim C$.
    These three properties are what is meant by saying that $\sim$ is an \emph{equivalence relation}.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item The identity function $f(x) = x$ is a bijection
  \item If $f : A 	o B$ is bijective then $f^{-1} : B 	o A$ is bijective.
  \item Let $f : A 	o B$ and $g : B 	o C$, since $g \circ f : A 	o C$ is bijective we have $A \sim C$.
   \end{enumerate}

Exercise 1.5.5

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Exercise 1.5.5

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