MATH1081 Test S1 v1a

Transcription

1 MATH08 Test 008 S va February 5, 05 These solutios were writte by Gary Liag ad typed up by Breda Trih. Please be ethical with this resource. It is for the use of MathSOC members, so do ot repost it o other forums or groups without askig for permissio. If you appreciate this resource, please cosider supportig us by comig to our evets ad buyig our T-shirts! Also, happy studyig :) We caot guaratee that our workig is correct, or that it would obtai full marks - please otify us of ay errors or typos at uswmathsoc@gmail.com, or o our Facebook page. There are sometimes multiple methods of solvig the same questio. Remember that i the real class test, you will be expected to explai your steps ad workig out.. Firstly, defie A : {0k k Z} ad B : {5m 8 m Z}. Let x A. Therefore, x 0k where k Z. Maipulatig this expressio, x 0k 0k (5k 3) 8 5m 8 (where k 3 m Z) x B A B To prove it is a proper subset, we eed to fid a elemet i B ot i A. Choose m 0 ad so we have 8 B. Seeig if this elemet is i A, cosider 0k 8 k 3 / Z. Thus, A B.

2 . Sice g f is oe-to-oe, we have g f (x ) g f (x ) x x. We eed to show that give the above is true, f (x ) f (x ) x x. Cosiderig the LHS, f (x ) f (x ) g (f (x )) g (f (x )) g f (x ) g f (x ) (sice g f is oe-to-oe) x x f (x ) f (x ) x x Therefore, f is oe-to-oe. 3. Combiig the two terms o the left ad simplifyig, LHS (k ) (k ) (k ) (k ) (k ) (k ) ((k ) (k )) ((k ) (k )) [(k ) (k )] (k k ) (k k ) (k ) (k) (k ) k RHS (k ) for k >. (differece of two squares)

3 Simplifyig the sum usig the prove result, k k (k ) k (k ) k (k ) (k ) k " # (k ) (k ) k k " # (k ) k (k ) k0 " # (k ) k (k ) ( ) k ( ) 5. ( ) 3

4 MATH08 Test 008 S vb February 5, 05 These solutios were writte by Joha Blaco ad typed up by Georgia Tsambos. Please be ethical with this resource. It is for the use of MathSOC members, so do ot repost it o other forums or groups without askig for permissio. If you appreciate this resource, please cosider supportig us by comig to our evets ad buyig our T-shirts! Also, happy studyig :) We caot guaratee that our workig is correct, or that it would obtai full marks - please otify us of ay errors or typos at uswmathsoc@gmail.com, or o our Facebook page. There are sometimes multiple methods of solvig the same questio. Remember that i the real class test, you will be expected to explai your steps ad workig out.. Let A {a}, A A {A }, A 3 A {A }. (i) A 3 A {A } A {A } {A {A }} {a} {{a}} {{a} {{a}}} {a, {a}} {{a, {a}}} {a, {a}, {a, {a}}} So the elemets of A 3 are a, {a} ad {a, {a}}. (ii) (i) {{a}} / A 3. False. {{a}} is ot i the list of elemets of A 3. (ii) {{a}} A 3. True, as {a} A 3 ad hece a possible subset is {{a}}.

5 . Sice g f is oe-to-oe, we have g f (x ) g f (x ) x x. We eed to show that give the above is true, f (x ) f (x ) x x. Cosiderig the LHS, f (x ) f (x ) g (f (x )) g (f (x )) g f (x ) g f (x ) (sice g f is oe-to-oe) x x f (x ) f (x ) x x Therefore, f is oe-to-oe. 3. Let k >. We have LHS 6 k k k (k ) (k ) 6 (k )(k ) k 8 6k 6 k k 8k 6k 6(k ) k(k ) 8k 6 k(k ). So 8k 6 k 3 k(k ) k(k ) k k 6 k k k (usig the prove result) k k k k 6 k k k! We shift the series i order to match the deomiators, k k3 k 6 k k k!

6 Next, we add or remove terms i order to match the limits of the sums, # " # " #! k k k " " k k k k! 9 6 k k k # k 9 k. 3

7 MATH08 Test 009 S vb February 8, 05 These aswers were writte by Joha Blaco ad typed up by Georgia Tsambos. Please be ethical with this resource. It is for the use of MathSOC members, so do ot repost it o other forums or groups without askig for permissio. If you appreciate this resource, please cosider supportig us by comig to our evets ad buyig our T-shirts! Also, happy studyig :) We caot guaratee that our aswers are correct - please otify us of ay errors or typos at uswmathsoc@gmail.com, or o our Facebook page. There are sometimes multiple methods of solvig the same questio. Remember that i the real class test, you will be expected to explai your steps ad workig out.. (i) To calculate this, recall that P (A) A. So A 3 8. (ii) (i) A A 3 : True. A 3 is the set of all possible subsets of of A, which icludes A. (ii) A A 3 : False. A set caot be a subset of its power set. You could write dow the set A to clearly show that A caot be a subset of A 3.. Rage (f) R (prove this). f is ot oe-to-oe sice it is either mootoically icreasig or decreasig (prove this). f is oto sice it s rage is the same as its codomai. f is ot a bijectio sice it is ot oe-to-oe (it must be both oe-to-oe ad oto to be bijective). 3. Combie all the fractios together o the LHS to obtai the RHS. Alteratively, split the RHS usig partial fractios. Usig the partial fractios result, ad creatig some telescopig sums, we obtai the aswer of 0 9.

8 MATH08 Test 009 S va February 5, 05 These solutios were writte by Gary Liag ad typed up by Breda Trih. Please be ethical with this resource. It is for the use of MathSOC members, so do ot repost it o other forums or groups without askig for permissio. If you appreciate this resource, please cosider supportig us by comig to our evets ad buyig our T-shirts! Also, happy studyig :) We caot guaratee that our workig is correct, or that it would obtai full marks - please otify us of ay errors or typos at uswmathsoc@gmail.com, or o our Facebook page. There are sometimes multiple methods of solvig the same questio. Remember that i the real class test, you will be expected to explai your steps ad workig out.. (i) P (A) {φ, {}, {s}, {w}, {, s}, {, w}, {s, w}, A} (ii) A P (A) P (A P (A)) 33 (A ad P (A) are disjoit).. (i) It meas that f (x ) f (x ) for x, x A implies that x x. Similarly, if g is a oe-to-oe fuctio, g (b ) g (b ) b b. I other words, every value i the codomai has at most oe iput value (value i the domai) that maps to it. (ii) f (x ) f (x ) x x g (y ) g (y ) y y.

9 Cosider g f (x ) g f (x ) g (f (x )) g (f (x )) f (x ) f (x ) (sice g is oe-to-oe) x x. (sice f is oe-to-oe) 3. Wheever simplifyig these, always state whatever law you have used i your workig. (A B c ) (Ac B c )c (A B c ) ((Ac )c (B c )c ) c (A B ) (A B) c (De Morga s Law) (Double complemet) [(A B ) A] B (Associative Law) A B. (Absorptio Law)

10 MATH08 Test 00 S vb February 8, 05 These aswers were writte by Joha Blaco ad typed up by Georgia Tsambos. Please be ethical with this resource. It is for the use of MathSOC members, so do ot repost it o other forums or groups without askig for permissio. If you appreciate this resource, please cosider supportig us by comig to our evets ad buyig our T-shirts! Also, happy studyig :) We caot guaratee that our aswers are correct - please otify us of ay errors or typos at uswmathsoc@gmail.com, or o our Facebook page. There are sometimes multiple methods of solvig the same questio. Remember that i the real class test, you will be expected to explai your steps ad workig out.. Let A {a}, B {b, c}, C {d, e, f, g, h, i, j}. (i) A B {(a, b), (a, c)}. So P(A B) {{(a, b)}, {(a, c},, {(a, b), (a, c)}}. (ii) B C as B, C. The P(B C).. Let f, g : R R with f(x) x, g(x) e x. (i) (f g) f(g(x)) (e x ) e x. (g f)(x) g(f(x)) e x. (ii) f g is oe-to-oe (prove this). g f is ot oe-to-oe (prove this). 3. By ispectig the give formula ad the result we must prove, we see it is reasoable to let A k ad B k ad rearrage to obtai the aswer.

11 Usig the proved formula to simplify the sum, we see this is a telescopig sum which simplifies to ta k ta (k ) k ta. ta