Fundamental Concepts of Mathematics

ANSWERS TO EXERCISES

ANSWERS TO EXERCISES 1.5 1. All 2. (a), (c) and (d) 3. F C A, 8 C D, Eec 4. (a), (d) 5. A =et>, 8 = {et>} and C= {et>, {et>}} 6. et>, {et>}, {l/>, {et> H, {et>, {et>}, {et>, {et>}}} 7. {a,b,e}, {a,b}, {a,e}, {b,e}, {a}, {b}, {e}, et> 8. None 9. A, {O}, {{ 1, 2 H, l/> 10. All 1.9 1. (a) {a, b, e, d, e}, {a, e, d, e}, {a, b, e, d, e} (b) {e, d}, {e}, {a} 2. (a) C, E (b) D, E (c) A, 8, D (d) None 3. (b), (c), (d) 4. (a), (b), (c) 5. Not always true; consider a =e 1= b 7. A = {o, 1}, 8 = {1} and C= {O} 13. All are false 16. (b) {{1, 2}, {2, 3, 4}, {4H 17. (a) neither (b) subset (c) element (d) subset and element (e) element 24. (b) (A IA) I (818) 25. (a) At nA 2 n ... nA; (b) Tt =A t, T;=A;\(A t UA2 U ... UA;_t) (i=2,3, ... ,n) 1.11 1. {(1,a), (2, a), (3,a), (1,b), (2, b), (3,b)}, {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3) } 3. (a, a) 4. A = C = {1 }, 8 = D = {2} 244 Fundamental Concepts of Mathematics 1.14 1. (a) Yes (b) No (c) No 2. (a) Yes (b) Yes 10. No; no 20. (b) No 2.4 2.9 1. (a) Yes; -1 (b) Yes; 2 (c) No 6. (a) n(n + 1) 2 (b) a a (1 _a)2 , 1 -a ' - (1 - a)2 3.2 1. 90 2. 24 3. 2 4. 5880; 3000 5. (2n - 1) X (2n - 3) X ... X 1 3.5 1. 360 2. 2880 3. 3456 4. 1440 5. (n - 1)! (n - 2) 6. (n - 2)! (n2 - 3n +3) 7. (n - 2)! (n - 3)(n + 2) 3.7 1. 125 2. 1024 3. (a) 648 (b) 320 4. 256;232 n(n r -1) 5. ---:-n -1 Answers to Exercises 7. 6198 n 8. (a) 2n - 22 if n is even; n+l 2n - 2-r if n is odd n (b) ~ X if if n is even; . n+l n-1 - -2- X 2 2 if n is odd 3.10 1. 126 2. (a) 700 (b) 340 (c) 174 3. (m + 1)! n!(m + 1 -n)! 4. (a) 144 (b) 12 5. 144 6. 360 7. 30 8. 3600 3.12 1. 35 2. (i) 84 (ii) 84 3. (i) 330 (ii) 462 4. (i) 2002 (ii) 6 (iii) 1876 5. 2024 6. (mn)! (ffl!)n n! 7. (a) 225 (b) 465 (c) 240 245 246 3.14 3.16 Fundamental Concepts of Mathematics 8. 26; 52 9. (a) 10 (b) 10 (c) 45 (d) 6 10. 5760 11. 2520 12. 2366 (m + 2)(m + l)(n + 2)(n + 1) 13. 4 14. 502 16 n!' . r! s! (n - r - s)! n! 17'~bl I a. . c. 18. n(n - :)(n - 5) (n - 3)(n2 -- 9n + 26) 19. 6 21. 36 1. 2n - 1 2. 300 3. 875 4. 15 8. 2001 n(n + 1) 6. (a) 2 4 1 13. (a) x4 + 4x2 + 6 + - + 4" x x (b) 32xs - 40Ox4 + 2000x3 - 10000x2 + 6250x - 3125 14. -10 15. v:16 . n = 16, a = 8 17. 1 + 8x + 36x2 + 104x3 + ... 18. -2; -345 Answers to Exercises 19. -576 21. a =±2, b =±1 22. m = 3, n = 2 23. (b) (i) (xy)2 - xy - 420 = 0 (iii) x = 5, y =-4 24. x = 3, a = 2, n = 6 25. y7 _ t y s + 14y3 - 7y 26. a = 7, b = 14, c = 7 27. (a) 3-3x+9x2 -15x3 + ... 1 1 (b) --