Answer Happy

3. Are the following collections necessarily partitions of the indicated sets? Justify your answers. (a) Let S be the set of students at a university, and let 6 = (Bo, B1, B2, B3. B4, Bs) where Bo = the set of students with no credit cards, B; = the set of students with exactly one credit card, B2 = the set of students with exactly two credit cards, B; = the set of students with exactly three credit cards, Be = the set of students with exactly four credit cards, Bs = the set of students with five or more credit cards. (b) Let B be the set of books in the Library of Congress, and let 8 = {S.L), where S is the set of books in the Library of Congress with fewer than 200 pages and L is the set of books in the Library of Congress with 200 or more pages. (c) Let A be the set of dresses, and let 6 = [S, D,P), where S = the set of dresses with stripes, D = the set of dresses with dots, P = the set of dresses with no stripes. (d) Let ST be the set of 50 U.S. states, and let C = (A, B, C.... Z), where A = the set of states starting with the letter A, B = the set of states starting with the letter B. and so on. (e) Let O be the set of quadrilaterals, and let 9 = {R, S.T. O), where R = the set of rectangles, S = the set of squares, T = the set of trapezoids, 0 = Q(RUSUT). 4. For m e N. let C = {x € Rm-1 < x < m). Is C = {C.Ime N) a partition of R? 5. Determine which of the following collections form a partition of R2. (a) For b € R, let lo = {(x, y) € Rly = b), and let 6 = { lb € R). (b) Form € R, let L, = f(x,y) € R2 y = mx), and let 9 = {Lmm € R). (c) For 1 > 0. let S, = {(x, y) € Rº max{1xl.lyl) = 1), and let & = {SIT E [0, 0)). (d) Forr > 0. let C = {(x, y) € R2x2 + y2 = r?), and let