Basic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.

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1 Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the cardiality of the set, E, where E is defied as E = {x R : si(x) = 1/2 ad x 5} Example 3. Suppose A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7} ad C = {2, 8, 4}. Fid: (a) A B (b) A \ C (c) B \ A (d) B C (e) C \ B Example 4. Prove that {9 : Z} {3 : Z}, but {9 : Z} {3 : Z}. Example 5. Prove that {9 : Q} = {3 : Q}. Fuctios Example 6. For each of the followig, determie the largest set A R, such that f : A R defies a fuctio. Next, determie the rage, f(a) := {y R : f(x) = y, for some x A}. (a) f(x) = 1 + x 2, (b) f(x) = 1 1 x, (c) f(x) = 3x 1, (d) f(x) = x 3 8, (e) f(x) = x. x 3 MSU 1

2 Ijective, Surjective, Bijective Fuctios Example 7. A fuctio f : Z Z Z is defied as f((m, )) = 2 4m. Verify whether this fuctio is ijective ad whether it is surjective. Example 8. Defie the operatio f(p) := d dx p. Does f defie a fuctio from P 4 to P 4? Justify your aswer. Is f a ijective fuctio from P 4 to P 4? Justify your aswer. Is f a surjective fuctio from P 4 to P 4? Justify your aswer. Example 9. Prove that the fuctio f : R \ {2} R \ {5} defied by f(x) = 5x+1 x 2 is bijective. Example 10. Prove or disprove that the fuctio f : R R defied by f(x) = x 3 x is ijective. Hit: A graph ca help, but a graph is ot a proof. Example 11. Let A = R \ {1} ad defie f : A A by f(x) = (i) Prove that f is bijective. (ii) Determie f 1. Example 12. The fuctio f : R 2 R 2 is defied by (a) Show that f is a bijectio. (b) Determie the iverse f 1 of f. f(x, y) = (2x 3y, x + 1) Example 13. Defie the fuctio, f : P 3 R via the operatio f(p) := 1 0 p(x)dx. Is f ijective ad or surjective from P 3 to R? Justify your aswer. x x 1 for all x A. Example 14. Cosider the fuctio f : R N N R defied as f(x, y) = (y, 3xy). Check that this is bijective; fid its iverse. Carefully justify that your aswer does ideed yield the iverse fuctio. Geeral Math Proofs Example 15. Assume that you kow that x < y. Carefully justify the statemet that x < x + y < y. 2 Example 16. Suppose a, b R. If a is ratioal ad ab is irratioal, the b is irratioal. Example 17. Show that there exists a positive eve iteger m such that for every positive iteger, 1 m Example 18. Prove: For every real umber x [0, π/2], we have si x + cos x 1. Example 19. Suppose x, y R +. Prove if xy > 100 the x > 10 or y > 10. MSU 2

3 Logic Example 20. For the sets A = {1, 2,..., 10} ad B = {2, 4, 6, 9, 12, 25}, cosider the statemets P : A B. Q : A \ B = 6. Determie which of the followig statemets are true. (a) P Q (b) P ( Q) (c) P Q (d) ( P ) Q (e) ( P ) ( Q). Example 21. Let P : 15 is odd. ad Q : 21 is prime. State each of the followig i words, ad determie whether they are true or false. (a) P Q (b) P Q (c) ( P ) Q (d) P ( Q) Example 22. Rewrite the followig usig logical coectives ad quatifiers (a) If f is a polyomial ad its degree is greater tha 2, the f is ot costat. (b) The umber x is positive but the umber y is ot positive. Example 23. Which of the followig best idetifies f : R R as a costat fuctio, where x ad y are real umbers. (a) x, y, f(x) = y. (b) x, y, f(x) = y. (c) y, x, f(x) = y. (d) y, x, f(x) = y. Example 24. Negate the followig statemets: (a) y Z, x Z, x + y = 1. (b) x Z, y Z, xy = x. Example 25. I each of the followig cases explai what is meat by the statemet ad decide whether it is true or false. (a) lim x c f(x) = L if ε > 0 δ > 0 such that 0 < x c < δ f(x) L < ε. (b) lim x c f(x) = L if δ > 0 ε > 0 such that 0 < x c < δ f(x) L < ε. (c) f : A B is surjective provided y B, x A such that f(x) = y. MSU 3

4 Iductio Example 26. If is a o-egative iteger, use mathematical iductio to show that 5 ( 5 ). Example 27. Prove by iductio that i 2 = ( + 1)(2 + 1). 6 Example 28. Prove that if N, the is divisible by 25. Eve Odd Proofs i=1 Example 29 (Prove usig Cotradictio). Suppose a Z. Prove that if a 2 is eve, the a is eve. Example 30. Prove that If a, b Z, the a 2 4b 2. Example 31. Let be a iteger. Show that 2 is odd if ad oly if is odd. Example 32. Suppose a, b, c Z. If a 2 + b 2 = c 2, the a or b is eve. Example 33. Prove that there is o largest eve iteger. Example 34. Prove the followig claim Claim: Suppose a Z. If a 2 2a + 7 is eve, the a is odd. Real Aalysis Idexed Sets Example 35. Let B 1 = {1, 2}, B 2 = {2, 3},..., B 10 = {10, 11}; that is, B i = {i, i + 1} for some i = 1, 2,..., 10. Determie the followig: (i) (iii) (v) (vii) 5 i=1 7 B i B i (ii) (iv) 10 B i i=1 i=3 i=j 10 i=1 j+1 B i (vi) B i B i+1 B i, where 1 j < 10 i=j i=j k B i, where 1 j k 10. k (viii) B i, where 1 j k 10. Example 36. Prove that x N[3 (1/x) 2, 5 + (1/x) 2 ] = [3, 5]. MSU 4

5 Bouded ad Ubouded Sets Example 37. Discuss whether the followig sets are bouded or ot bouded. (a) A = { 2, 1, 1/2}. (b) B = (, 2). (c) C = {1/2, 3/2, 5/2, 7/2, 9/2,... } = { 2 1 N}. 2 { } (d) D = ( 1) : N (e) E = { 1 : Q \ {0} } Sequeces Example 38. For each of the followig, determie whether or ot they coverge. If they coverge, what is their limit? No proofs are ecessary, but provide some algebraic justificatio. (a) { } N (b) { si( π 4 )} N (c) {(1 + 1/) 2 } N (d) {( 1) } N (e) { } N Example 39. Usig the defiitio of covergece, that is, a ε N argumet, prove that the followig sequeces coverge to the idicated umber: 1 (a) lim = 0. 1 (b) lim = 0. (c) lim = 1 2. Example 40. For each of the followig, determie whether or ot they coverge. If they coverge, what is their limit? No proofs are ecessary, but provide some algebraic justificatio. { } (a) 3 + ( 1) 2 N { } (b) (c) { } +1 N N\{1} MSU 5

6 Example 41. Usig the defiitio of covergece, that is, a ε N argumet, prove that the followig sequeces coverge to the idicated umber: (a) lim (3 + 2 ) = 3. 2 (b) lim si() = 0. Example 42. Prove that the sequece {( 1) } N does ot coverge. Example 43. Use the formal defiitio of the limit of a sequece to prove that 2 1 lim = 2 3. Ope ad Closed Sets Example 44. Which of the followig sets are ope? (i)( 3, 3) (ii)( 4, 5] (iii)(0, ) (iv){(x, y) R 2 : y > 0} (iv) 5 =1 ( 1 + 1, 1 1 ), where N (v){x R : x 1 < 2}. Example 45. Let A = [2, 5]. Discuss whether A c a ope set. Example 46. Discuss which of the followig sets are ope. (a) (2, 3). (b) ( 4, 8]. (c) [1, 3). (d) (, ). (e) [1, 5] [2, 3]. Example 47. Which of the followig sets are closed? Justify your aswer. (i) A = [2, 5] (ii) B = ( 1, 0) (0, 1) (iii) C = {x R : x 1 < 2} (iv) D = { 2, 1, 0, 1, 2} (v) Z MSU 6

7 Liear Algebra Vector Spaces Example 48. Show that the set R 2 over R is ot a vector space uder the followig defiitios for vector additio ad scalar multiplicatio: x + y := (x 1 y 1, x 2 y 2 ) ad λx := (λx 1, λx 2 ), where x = (x 1, x 2 ) R 2, y = (y 1, y 2 ) R 2, ad λ R. Example 49. Uder the usual matrix operatios, is the set {( ) } a 0 a, b, c R b c a vector space over R? Justify your aswer. Example 50. Defie the set V = {(x 1, x 2 ) R 2 : x 2 2x 1 + 1}. Sketch a picture of the set V iside of the plae, R 2. Is V a vector space over R? Justify your aswer! Example 51. Defie Is V a vector space over R? V = {p P 2 : x R, p (1) = 0}. Example 52. Fid the additive iverse, i the vector space, of the followig: (a) I P 3, of the elemet 3 2x + x 2. (b) I the space of 2 2 matrices, of the elemet ( ) (c) I {ae x + be x a, b R}, the space of fuctios of the real variable x, the elemet 3e x 2e x. You may assume that these vector spaces are defied over R ad that i each case, atural defiitios of additio ad scalar multiplicatio hold. Example 53. Show that the set of liear polyomials P 1 = {a 0 +a 1 x a0, a 1 R} uder the usual polyomial additio ad scalar multiplicatio operatios is a vector space over R. Liear Maps Example 54. Defie the fuctio A : P 3 P 5 by Is A a liear fuctio? Justify your aswer. A(p)(x) = x 2 p(x) for x R. MSU 7

8 Example 55. Suppose b, c R. Defie T : R 3 R 2 by Show that T is liear if ad oly if b = c = 0. T (x, y, z) = (2x 4y + 3z + b, 6x + cxyz). Example 56. For each of the followig L, aswer yes or o, ad briefly justify your aswer: (a) Is L : R R, with L(x) = si(x), a liear fuctio? (b) Is L : R R, with L(x) = x 1/2, a liear fuctio? (c) Is L : R R, with L(x) = 51.5x, a liear fuctio? (d) Is L : P 3 P 3 defied by L(p) = 3p a liear fuctio? Fid the images uder L of p, q P 3 defied by p(x) = x 3 7 ad q(x) = 2x 2 + 3x + 5. Example 57. Defie the fuctio T : R 2 R 2 by the trasformatio T ((x 0, x 1 )) = (x 0, 0), where (x 0, x 1 ) R 2. Is T a liear fuctio? Justify your aswer. Abstract Algebra Divisibility ad Remaiders Example 58. Use the Euclidea Algorithm to fid the greatest commo divisor for each of the followig pairs of itegers: (a) 51 ad 288 (b) 357 ad 629 (c) 180 ad 252. Example 59. Prove that the square of every odd iteger is of the form 4k + 1, where k Z (that is, for each odd iteger a Z, there exists k Z such that a 2 = 4k + 1). Example 60. Prove that if a divides b ad c divides d, the ac divides bd. Example 61. Aswer true or false ad give a complete justificatio. If p is prime, the p is prime. Example 62. Let a, b, c Z. Prove that if gcd(a, b) = 1 ad c b the gcd(a, c) = 1. (Hit: Use proof by cotradictio) MSU 8

9 Equivalece Relatios ad Modular Arithmetic Example 63. For (a, b), (c, d) R 2 defie (a, b) (c, d) to mea that 2a b = 2c d. Show that is a equivalece relatio o R 2. Example 64. Defie a relatio o Z as x y if ad oly if 4 (x + 3y). Prove is a equivalece relatio. Describe its equivalece classes. Example 65. Let X = R 2, the xy-plae. Defie (x 1, y 1 ) (x 2, y 2 ) to mea x y 2 1 = x y 2 2. Is a equivalece relatio? Justify your aswer. Give a geometric iterpretatio of the equivalece classes of. Example 66. Do the followig calculatios i Z 9 (see page 238 of the text for a descriptio of this otatio), i each case expressig your aswer as [a] with 0 a 8. (a) [8] + [8] (b) [24] + [11] (c) [21] [15] (d) [8] [8]. Example 67. Let a ad b be give itegers. Prove a b mod 5 if ad oly if 9a + b 0 mod 5. MSU 9