1. Using the axioms for a group determine which of the following are groups. State your reasoning (i) The set Q+ = {q = Q: q>0} under ×. (ii) The set Q+ = {q € Q : q > 0} under division. (iii) The set {1, 2, 3, 4, 5, 6, 7} under × mod 8. (iv) The set G = {( а -b b a b ) | ab € R ,bЄ R with a and b not both under matrix multiplication.

(1.) Let T = {1, 2, 3, 4, 5, 6, 7, 8, 9}.  Suppose five integers are chosen from T. Show that it is not true that there must be two integers whose sum is 10 by giving a counterexample. In other words, fill in the blank with five numbers from T, no two of which have a sum of 10. (Enter your answer in set-roster notation.) (2.) Suppose five pairs of similar-looking boots are thrown together in a pile. What is the minimum number of individual boots that you must pick to be sure of getting a matched pair? Why? Since there are 5 pairs of boots in the pile, if at most one boot is chosen from each pair, the maximum number of boots chosen would be............? . It follows that if a minimum of............. boots is chosen, at least two must be from the same pair.

Set notation

Find the adjoint of