91Ó°ÊÓ

Theorem 2states 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\(S = {\mathop{\rm aff}\nolimits} S\).

Consider\(A\)as a line segment through the point\(\left( {0,0} \right)\)to\(\left( {1,0} \right)\)shown below:

\(A = \left\{ {\left( {{\mathop{\rm x}\nolimits} ,y} \right)|y = 0,0 \le {\mathop{\rm x}\nolimits} \le 1} \right\}\)

Consider B as the line segment from the point\(\left( {1,0} \right)\)to\(\left( {2,0} \right)\)shown below:

\(B = \left\{ {\left( {{\mathop{\rm x}\nolimits} ,y} \right)|y = 0,1 \le x \le 2} \right\}\)

Thus, the intersection of\(A\)and\(B\)is only a single point. Therefore it is shown below:

\(\begin{aligned}{}{\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) = A \cap B\\ = \left\{ {\left( {1,0} \right)} \right\}\end{aligned}\)

A line passing through the origin and the point\(\left( {1,0} \right)\)is an affine subset of A, and a line passing through the point\(\left( {1,0} \right)\)and\(\left( {2,0} \right)\)is an affine subset of B. Therefore,\({\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B\)is a line, which represents\({\mathop{\rm x}\nolimits} - \)axis because the first set is a point and the second is a line. Hence, the\({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)does not hold.

Thus, it is proved that equality does not hold.