Media not available in the Digital Archive
Description
This programme explains the idea of classification using equivalence relations using examples from geometry.
Module code and title: M101, Mathematics: a foundation course M101; 29; 1986 14-09-1986 1986 00:24:45 + Show more... David Saunders Bob Coates; Allan I.,1936-2013 Solomon; Bob Margolis BBC Open University Area; Euclidean congruences; Geometry; Hyperbolas; Matrix transformations; Triangles This programme explains the idea of classification using equivalence relations (introduced in section 2 of the unit) using examples from geometry. The programme starts by posing a problem involving the areas of two triangles; although there is a straightforward but lengthy way of solving the problems using calculation, the ideas about to be introduced provide a much simpler solution. These ideas are explained by looking first at the symmetries of a given equilateral triangle, and asking how the points of the plane can be classified using these symmetries. A computer animation shows that in general there are sets containing six equivalent points, where each point is related to any other by either a reflection or a rotation which preserves the triangle. The programme explains that such a relation is characterised by the properties of being symmetric, reflexive, and transitive, and is called an equivalence relation. Students are then reminded of a problem involving matrices from the television section of the text, and are shown the strategy for proving that the relation so defined is actually an equivalence relation; computer animation demonstrates that the corresponding sets of equivalent points are concentric circles. It is explained that the matrix transformations - rotations - are special cases of Euclidean congruencies, and in fact a rotation can be used to give a simple solution to the problem from the beginning of the programme. Finally, students are posed a further problem involving matrix transformations which give rise to hyperbolas rather than circles. HOU5543 FOUM258K 2491 no