Answer Happy

1. [10 points] Give all appropriate answers to the questions below. (a) Every opposite symmetry of a tetrahedron (with centroid/center-of-mass at the origin of R³) is a direct symmetry of the tetrahedron composed with multiplication of all coor- dinates by -1. (A) True (B) False (b) A permutation which swaps two elements of a set and leaves all others in place is called a: (A) transposition (B) 2-cycle (C) (2,2)-cycle (D) transmogrification (c) Consider the bijection f: {1,2,...,7}{1,2,...,7} defined by the values f(1) =2, f(2)=3, f(3) =6, f(4)=1, f(5)=7, f(6)=4, f(7) = 5. As a permutation, f is (A) even (B) odd (d) Twists and translations of R³ are both: (A) opposite with no fixed points (B) direct with at least one fixed point (C) direct with no fixed points (D) opposite with at least one fixed point (e) Unlike the situation in R², the dihedral group D₁ can be realized as a group of direct symmetries in R³. (A) True (B) False