(5) (22 Points) Consider the relation =1 on the set R given by (Vx € R)(WY ER) X =1y A3z e Z such that y - x = z. Equiva

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5 22 Points Consider The Relation 1 On The Set R Given By Vx R Wy Er X 1y A3z E Z Such That Y X Z Equiva 1 (120.33 KiB) Viewed 22 times

(5) (22 Points) Consider the relation =1 on the set R given by (Vx € R)(WY ER) X =1y A3z e Z such that y - x = z. Equivalently, (Vx € R)(VY ER) X =1 y + y - x E Z. (a) Prove that =1 is an equivalence relation on R. (b) (i) Find 3 elements in the equivalence class [0]. (ii) Find 3 elements in the equivalence class [?]. (c) (i) Give the definition of a binary operation on a set X. (ii) Explain why multiplication is a binary operation on R. (iii) If we let R/ ~ denote the set of all equivalence classes with respect to the =ı relation on R, explain why multiplication on equivalence classes defined as [2] * [y] = [2 * y) does not define a binary operation on R/ ~. Hint: You may find it useful to consider [0] * [1/2] and [1] * [1/2]. =