EDB — 092

Exercises

1. [092] Let \(D,C\) be non-empty sets and \(f:D→ C\) a function. Let \(I\) be a non-empty family of indexes, \(A_ i⊆ D\), for \(i∈ I\). Given \(A⊆ D\) remember that the image of \(A\) is the subset \(f(A)\) of \(D\) given by

   \[ f(A){\stackrel{.}{=}}\{ f(x), x ∈ A\} ~ ~ . \]

   Show these image properties.

   \begin{eqnarray} f\bigl(⋃_{i∈ I} A_ i \bigr) & =& ⋃_{i∈ I} f\bigl(A_ i \bigr) \nonumber \\ f\bigl(⋂_{i∈ I} A_ i \bigr) & ⊆ & ⋂_{i∈ I} f\bigl(A_ i \bigr) \nonumber ~ ~ . \end{eqnarray}

   Show that the function is injective if and only if

   \begin{equation} f\bigl( A_ 1∩ A_ 2 \bigr) = f\bigl(A_ 1 \bigr) ∩ f\bigl(A_ 2 \bigr) \end{equation}

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   is an equality for every choice of \(A_ 1,A_ 2⊆ D\).