## Set Theory for the Mathematician (Holden-Day Series in by Jean E Rubin

By Jean E Rubin

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C D Find all ten symmetries of P, describing them geometrically and in permutation notation. (b) Work out the effect of the two possible compositions of reflection in the line OA with reflection in the line OC. (c) Work out the effect of the two possible compositions of reflection in the line OA with rotation through 3/5 of a turn anti-clockwise. 5. Determine the symmetry groups of each of the following plane figures. (i) v ◦ rrr vv rr v v rr vv rr v v ◦A i◦ i i iiiii AA ØTiTTT !! Ø AA !

Pattern cm. The fundamental region has a reflection (Sx | 0) in the x-axis and the holohedry group is {(I | 0), (Sx | 0)}. There are glide reflections (Sx | u) and (Sx | v) in lines parallel to the x-axis and which compose to give the translation (I | u + v). The whole symmetry group is {(I | mu + nu) : m, n ∈ Z} ∪ {(Sx | mu + nv) : m, n ∈ Z}. 8. 22. 23. p6 Pattern cmm. The fundamental region has reflections in the x and y-axes as well as the half rotation about the origin. So the holohedry group is {(I | 0), (−I | 0), (Sx | 0), (Sy | 0)}.

W1 w2 w3 Proof. 3) u1 u2 u3 (u × v) · w = v1 v2 v3 . w1 w2 w3 This quantity is often called the vector triple product of u, v, w and written [u, v, w]. 2. 4. Find implicit and parametric equations for the plane P containing the points with position vectors p = (1, 0, 1), q = (1, 1, 1) and r = (0, 1, 0). Solution. Let us begin with a parametric equation. Notice that the vectors u = q − p = (0, 1, 0), v = r − p = (−1, 1, −1) are parallel to P and linearly independent since neither is a scalar multiple of the other.