Question

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

23. \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\). (To show that \(D \cup E \subset F\), show that \(D \subset F\) and \(E \subset F\).)

Short answer

It is proved that \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

Step-by-step solution

Set S is affine

RecallTheorem 2,whichstates that a set \(S\) is affineif and only if every affine combination of points of \(S\) lies in \(S\).

That is, \(S\) is affine if and only if \(S = {\mathop{\rm aff}\nolimits} S\).

Show that \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\)

Since \(A \subset \left( {A \cup B} \right)\)so, it follows that \({\mathop{\rm aff}\nolimits} A \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\) according to exercise 22. Likewise, \({\mathop{\rm aff}\nolimits} B \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

Therefore, \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).

Hence, it is proved that \(\left( {\left( {{\mathop{\rm aff}\nolimits} A} \right) \cup \left( {{\mathop{\rm aff}\nolimits} B} \right)} \right) \subset {\mathop{\rm aff}\nolimits} \left( {A \cup B} \right)\).