Week 1 - Basic Set Theory

Friday, 15 February 2019

10:58 PM

Machine generated alternative text: Consider the three individual elements 1 , 1 and 2. If we consider these elements as a single unordered collection of distinct objects then we call it the set {1, 1 , 2}. Because sets are unordered this is the same as {2, 1, 1} and because we only collect distinct objects this is also the same as {1, 2}. For example, let A — {3, 2, 4, 4} , B = {2, 4, 1, 1, 3} and C — {1, 3, 2, 4, 2, 1}. a) If every element of the set A is also an element of the set Bthen we say that A is a subset of B: A C B. Which sets are subsets of one another? ZACB ZCCB ZBcc Z ACC Submit part You chose a correct answer. You were awarded 1 mark. You chose an incorrect answer. You chose a correct answer. You were awarded 1 mark. You chose a correct answer. You were awarded 1 mark. You chose a correct answer. You were awarded 1 mark. You scored 4 marks for this part. Score: 4/4 Answered b) Sets are equal if they are subsets of each other. Which sets are equal? ZB=C DC—A Submit part You chose a correct answer. You were awarded 1 mark. You scored 1 mark forthis Dart.

 

Machine generated alternative text: Consider the sets A for {1, 2, 3}. a) {7, 6, 5, 3} and B What is the union A LJ B? set (1, 3, 4, s, 6, 7) b) {4, 1, 3}. Find the union, intsersection and set differences below. Use the NUMBAS syntax set(1,2,3) Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered What is the intersection A n B? set (3) Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

c) What is A — B, the set of elements from A which are not in B? Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered d) What is B — A, the set of elements from Bwhich are not in A? set (1, 4) Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

What is union of the set differences (A — B) IJ (B set (1, 4, s, 6, 7) f) A)? This is also called the symmetric difference AAB. Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Match the mathematical expression with its Venn diagram. An B AAB AUB o o o o o o o o o Score: 1/1 Answered o o o

 

Consider thesetsX = {2n + 91n e Z} and Y — {45k 111k €- Z}. a) Find an element of Y which is also an element of X. Submit part This is an elementofY. In fact Y C Xso any element ofYis automatically in X. You were awarded 1 mark. By the way, the easiest element to find is 11 which is obtained when k = 0. You scored 1 mark for this part. Score: 1/1 Answered b) Find an element of X which is not an element of Y. Submit part This is an element of X. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

c) The sets X and Y are not equal because OX c Y OYCX Submit part Yes. All you need to demonstrate is there is one element of X which is not also in Y. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

The powersetof A, written P(A), is the set of all subsets of A. What is the power set of {9, 8}? Show steps (Your score will not be affected.) Answer: set ( set ( ) , set (8) , set (9) , set (8, 9)) You revealed the steps. Your answer is numerically correct. You were awarded 1 mark. Because you received full marks for the part, your answers to the steps aren't counted. You scored 1 mark for this part. Score: 1/1 Answered

 

Machine generated alternative text: We use the notation IAI = n to mean that A contains n elements. This is often called the cardinality ofthe set. Here is a handy fact about the number of elements in a power set. If IAI = nthen = 2". Using this fact, answer the following questions regarding the power set. a) P(A) contains at least the elements { } and A. The case where = A is panicularly interesting. What is What is Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered Hence, what is Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

Machine generated alternative text: Hence is a set that only contains one element. But it must at least contain the element { Ponder this for a moment and then answer the question below, and remember that the NUMBAS syntax for { } is set() • (Your score will not be affected.) Hide steps Answer: set (set ( ) ) Submit part This feedback is based on your last submitted answer. Submit your changed answer to get updated feedback. You revealed the steps. Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Because you received full marks for the part, your answers to the steps aren 't counted. Because you received full marks for the part, your answers to the steps aren't counted. Score: 1/1 Answered

 

Machine generated alternative text: If ISI = 1 then what is Using the handy fact, = 21. Using this handy fact again, we deduce that = , which is Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered Use the handy fact yet again to determine (Your score will not be affected.) Hide steps 16 Answer. 16 Submit part You revealed the steps. Your answer is numerically correct. You were awarded 1 mark. Because you received full marks for the part, your answers to the steps aren't counted. You scored 1 mark for this part.

 

Machine generated alternative text: The Cartesian Product oftwo sets A and B is the set of all pairs (a, b) wherea e A and b €- B. Or, in the language of set theory: Ax B— e A,be B}. Consider thesets Aand Bwhere IAI = 5and IBI = 8. The cardinalityofA x Bis IA x = IAI x IBI Answerthe followingquestions using the NUMBAS synax for multiplication and A for exponent. For example 3*2A9 is the syntax for 3 x 29. a) Evaluate IA x 131. 40 Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered b) Evaluate IP(A x 2A (S*8) 25 x8 Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1

 

Machine generated alternative text: c) Evaluate IP(A) x rs * 25 x 28 Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

Machine generated alternative text: For two sets A, B the Cartesian product is Ax B— e A,be B}. A function f from A to B, written f : A B, is a subset of A x B subject to the condition that for all a e A there is exactly one b e B such that In otherwords, for every a the function must determine a unique output b. Since a determines b we usually indicate this with the function notation f(a) = b. For example, consider thesetsA = {2, 9, 8, 0} and B = {7, 8, 4} and f : A Bdefined by a) Which of the following are true? o Üf(0) — 8 Submit part You chose a correct answer. You were awarded 1 mark. You chose a correct answer. You were awarded 1 mark. You chose a correct answer. You were awarded 1 mark. You scored 3 marks for this part. Score: 3/3 Answered

 

Machine generated alternative text: b) For fto be a function there must be exactly one bvalue for every a value such that f(a) = b. This means that there are two ways that f can fail to be a function: either there are too many possible values of b, or too few. In this example f is nota function because there is more than one value of b such that f(2) = b. These are 02 09 no Submit part You chose a correct answer. You were awarded 0.5 marks. You chose a correct answer. You were awarded 0.5 marks. You scored 1 mark for this part. Score: 1/1 Answered

 

Machine generated alternative text: c) Here is a different reason why f is not a function becase there is one value ofa e A such that f(a) is not defined. This is Submit part You correctly identified that f(8) is undefined. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

 

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