Abstract Algebra

Solve #1 (a)-(d). Show ALL steps and proof.

GIVE ORIGINAL ANSWER!

1. [pts] Let (S, *) and (S', *') be any given two binary structures. Let o : S - S' be any given isomorphism from (S, *) to (S', *'). Let us show that the inverse map -1: S' - S exists, and is an isomorphism from (S', *') to (S, *) . (a) Define a map - from S' to S by o (s') = s if and only if o (s) = s'. Show that - is well-defined as a map, that is, [Vs' E S', Is E S, $ 1 (s') = $], that is, 's = (8) _q'S= E'S= ,SA us = IS + 28 = (8)_QVIs = (8)1-$'s= isA'ISA (b) Show that o-1 : S' - S is one-to-one, that is, (c) Show that o-1 : S' - S is onto, that is, [s = (8) 1 0',59 SE'S = $A (d) Show that o 1 : S' - S preserves the operations, that is