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.

In (Z24, €) find the orders of the elements 3, 4, 6, 7, and 18.

(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.

Create an example of a permutation and a combination (one of each) and use them to communicate the connection between permutations and combinations in a similar context with the same values. Use the example with the 13 ingredients for the soup as a guide. Your example should include a real life example different from the one presented but that connects permutations and combinations. Notice the example given used 13C3 and 13P3 and gave a reason why one was used over the other while still talking about soup ingredients.

Question 9 (Mandatory) (3.5 points) Saved (1) How many elements in Z₁13 and (2) what are possible orders in Z 13? a) (1) Number of elements: 12 and (2) possible orders: {1, 2, 3, 4, 6, 12} ○ b) (1) Number of elements: 13 and (2) possible orders: {1, 13} c) (1) Number of elements: 13 and (2) possible orders: {1, 2, 3, 4, 6, 12, 13} d) (1) Number of elements: 12 and (2) possible orders: {0, 1, 2, 3, 4, 6, 12} Previous Page Next Page Page 9 of 30