Sets and Relations Questions and Answers

green c yellow and d not red Tod plify your answer green plify your answer GELED GILE

The adjacency matrix below defines a directed graph with vertices A B C D E F G H 1 Write down the strong connected components as sets of vertices separated by commas and enclosed by braces 2 Give an example of one edge we can add to make it a strong connected graph Hint we recommend drawing the graph on your scrap paper 000 0 1 0 1 0 000 00000 100 0 0 0 000 0 1 01 O O O 1 101 00 O

Interpreting Venn diagram cardinalities with 2 sets for a real A college radio station surveyed 410 incoming freshmen to see how many like rap music and how many like classical music The Venn diagram below shows the results Each number gives the number of freshmen who fall into that Venn diagram category All freshmen in the survey Like rap 33 113 Like classical 22 242 a How many of the freshmen like classical music freshmen b How many of the freshmen like rap music or classical music or both freshmen c How many of the freshmen do not like both rap music and classical music freshmen

If A a b c d and B c d e f g then n A U B n a b c d e f g 7 n A 4 n B 5 and n An B n c d 2 Since 7 4 5 2 this suggests that n AUB n A n B n An B Use Venn diagrams to justify thi arbitrary finite sets A and B OA OB O C O D n AUB n AUB n AUB n AUB A A B A A B O O B n AUB B n ANB n ANB n ANB CALE

Let U be the set of natural numbers 1 through 20 let A be the set of even numbers in U and let B be the set of numbers in U that are divisible by 3 a Find An B AUB AUB ANB b What two pairs of the four sets of part a are equal a List all the elements of each set below ANB 1 2 3 4 5 7 8 9 10 11 13 14 15 16 17 19 20 Use a comma to separate answers as needed AUB 1 2 3 4 5 7 8 9 10 11 13 14 15 16 17 19 20 Use a comma to separate answers as needed AUB 1 5 7 11 13 17 19 Use a comma to separate answers as needed AnB 1 5 7 11 13 17 19 Use a comma to separate answers as needed b Select all the choices below that show two pairs of sets that are equal A ANB AUB C AUB AUB DE ANB AnB B AUB AnB D ANB AUB OF AUB ANB Points 0 2c

Determine if each of the following sets is closed under addition a P 0 c N 7 10 12 b S 0 7 14 21 28 d xlx E integers x 10 a P 0 is closed under addition False True

Consider the following zero coupon bonds Bond Yrs to Mat Yield to Maturity 0 0433 0 046 0 0495 0 0511 0 0531 0 0555 a b CE4 d e 123456 f 6 What is the one year forward rate starting at the end of year5 0 0636 O 0 0611 O 0 0654 O 0 0676

Problem 5 Let 0 1 For each of the following prove that the result holds for all sets X Y Z CE or provide a counterexample to disprove a XUY X UY b XnY X ny c X Y U Z XY U XZ

Find the domain and range of f 4 2 0 2 4 6 O Domain 4 2 6 8 7 5 0 Range 4 2 0 2 4 6 8 O Domain 5 8 Range 4 8 O Domain 4 8 Range 5 8 O Domain 4 8 Range 5 8 O Domain 5 8 Range 4 8 O Domain 4 2 0 2 4 6 8 f x 4 2 6 7 5 0

The following table shows the depth D in inches of snow on the ground h hours after the beginning of the nowfall h hours D depth 2 3 3 LO 5 n your response answer the following questions 1 What is the value of D 8 2 Calculate the average rate of change from 5 to 8 5 8 8 10

70 60 50 40 30 20 10 50 t y 100 150 290 t b From the equation identify the maximum point on the graph of y p t Round your answers to two decimal place X c In what year is the percent of women workers at its maximum according to this model x

Let s X ANB C and Y A B n A C 1 Draw a Venn diagram for sets X and Y 2 Determine which relationship C 2 between sets X and Y is true Show your work in a blue book

The set C contains 2 and 3 and is closed under addition a What whole numbers must be in C b What whole numbers may not be in C c Are there any whole numbers definitely not in C a Choose the correct set below that C must be O O The priime numbers 2 3 5 7 11 13 17 19 1 2 3 6 2 3 4 5 6 2 3 5

Finding sets and complements of sets For the universal set U 2 3 4 5 6 7 complete the parts below Write your answers in roster form or as 0 a Suppose A 2 5 7 Then what is A A 1 b Suppose we know that C 2 5 6 7 Then what would C have to be D

with 2 sets for a real world situation a writing course the professor surveyed the students to see the literature genres they like The Veendagram shows this information for two of the pres All students surveyed Like nonsense Rachel Like fantavy Donna a How many students like nonsense b How many students like nonsense or fantasy or both c which students do not like fantasy Choose all that apply Donne Abdul Isabel Bia Rachel x 5

Circular loops in a Venn diagram divide the universe U into distinct regions No sets 1 set 2 sets 2 1 region 2 regions Answer the following parts a through d OA 3 4 regions a Draw a diagram with three circles that gives the largest number of regions OB G OC

A person with the antigens A B or both A and B has the respective blood type A B or AB If neither antigen is present the type is O Draw and label a Venn diagram that illustrates the ABO system Choose the correct diagram below OA O C AB B 1 0 B CBD OB OD 0 OD A AB AB D

Let U a b c d e f g h A a b c f h B a c h and C a g h Locate all eight elements of U in a three loop Venn diagram and then list the elements in the following sets a BUC b AnB c BNC d AUC e C f ANC g AU BNC Locate all the elements of U in a Venn diagram Choose the correct answer below A U b C A f a O d g e C B OB A f a C B ORI C B b g A f O C d e

1 2 1 6 Let T R R be the linear transformation with standard matrix A 2 3 0 352 Show that T is one to one and find the standard matrix of T 1

Draw a Venn diagram and use the given information to fill in the number of elements in each region n A 15 n AnBnC 4 n An C 12 n An B 9 n B nC 9 n A nB n C 20 n B nC 5 n B UC 35 Complete the Venn diagram 0 A 0 E P B N WANY CITED

Union and intersection of finite sets Sets A and I are defined as follows A 1 4 5 I 2 1 0 3 4 8 Answer each part below Write your answer in roster form or as a Find the intersection of A and I AnI 0 b Find the union of A and I AUI 0 www X 010 8 0 0 0 5

C Give the following using the sets given in A and B above i Bn AUC ii C AI

5 A The Bursaries Committee of a certain country offers scholarships to only 4 students from a particular school who come out first second third and fourth in their final year examinations i Find the number of ordered ways the students can qualify for a scholarship if the school has 10 students sitting for their final examinations ii Solve for the problem i above if order is not important B i In a group of 100 people each person picks a number from 97 to 115 What is the minimum number of people you need to be sure two of them have the same number iii At a certain university 525 of the seniors are history majors or math majors or both There are 102 senior math majors and 35 seniors are majoring in both history and math How many seniors are majoring in history 10 Marks

Consider the following points 10 6 9 14 and 11 14 Step 2 of 2 What is the area of the triangle Hint Make use of the midpoint formula if the triangle is isosceles

Which is the better investment option Justify your answer with appropriate calculations Investment A Invest 100 each month for 3 years at 3 16 a compounded monthly Investment B Invest 300 each quarter for 3 years at 3 25 a compounded quarterly 7 11

1 point Which of the following functions are one to one Recall that injective is another word for one to one 1 2 3 1 2 3 4 A 2 8 to 21 Q B 7 d R R R B W B X 17 Y Z B X E Y Z B V W X Y Z 4 5 2 3 4 1 2 3 4 R R B W X 2 B W S X Y Z I

1 point Which of the following functions are bijective Recall that bigjective means that the function is both injective and surjectvie so both one to one and onto 1 2 5 R A 1 W 2 X F 3 Y 2 HB 8 1 2 3 4 31 to UL R R R R B B S X Y 2 B X Y B 1 3 2 3 B

WoodCom manufactures 4 types of wooden products Each type of product requires rocessing times for 2 different job shops Additionally a certain amount of wood is needed to roduce each type of product The resources required to make each type of product and the elling prices are given in the table below Currently 1200 minutes of job shop 1 time 800 inutes of job shop 2 time and 1000 m of wood available Formulate an LP that can be used o maximize revenue Job Shop 1 Time minutes Job Shop 2 Time minutes Wood m Selling Price S Product 1 2 3 4 8 18 14 20 2 2 6 80 6 8 4 2 12 20 18 40

omt P x is the statement x 3 4 Q x is the statement x 1 5 Q 6 V R 2 A True B False ving R x is the statement x 4 1 Determine the truth value of the following propositions P 1 Q 1 A True B False R 5 V P 1 A True B False cates

Let s be the following statement 10 1 point If it is raining then the ground is wet Which is the converse of s If it is not raining then the ground is not wet If it is raining then the ground is not wet If the ground is wet then it is raining If the ground is not wet then it is raining If the ground is not wet then it is not raining Which is the contrapositive of s If it is not raining then the ground is not wet If it is raining then the ground is not wet If the ground is wet then it is raining If the ground is not wet then it is raining If the ground is not wet then it is not raining Answer s submitted

8 1 point whether or not Complete the truth table and determ pVq p q PqpVq pvq p q p q TT TF FT FF Are the two statements equivalent A Yes the columns are identical B No the columns are not identical C No the rows are not identical D Yes the rows are identical

Let U Universal set 0 1 2 3 4 5 6 7 8 9 A 2 3 4 5 7 9 and B 2 6 7 9 List the elements of the following sets If there is more than one element write them separated by commas A AUB AUB AOR

Use the values in the following table 8 1 03 4 2 0 0 15 9 a Explain why the table defines y as a function of x For each value of y there are multiple values for x For some values of y there are multiple values for x O For each value of x there are multiple values for y For each value of y there is only one x For each value of x there is only one y b State the domain and range of this function Enter your answers as a comma separated list domain range X y 9 12 14 19 22 12 34 22 22 70 c If the table expresses y f x find f 0 and f 12 If the table does not express y f x enter DNE f 0 f 12

for the planets Jupiter and Saturn to determine the truth values of the simple statements Then determine the truth value of the compound statemen Phoebe has a larger diameter than Rhea if and only if Callisto may have water ice and Calypso has a diameter of 6 49 miles Click on the icon to display the information provided about the moons for the planets Jupiter and Saturn Determine the truth value of the first simple statement Phoebe has a larger diameter than Rhea Choose t True O False Determine the truth value of the second simple statement Callisto may have water ice Choose the correct False True Determine the truth value of the third simple statement Calypso has a diameter of 6 49 miles Choose the O True O False is the compound statement true or false O True O False Figure o Metis o Adrastra Amalthea o Thebe Jupiter OOOO lo Europa Ganymede Callisto o Leda Himalia O Lysithes o Elara o Ananke o Carme o Paiphae o Sinope Diameter of moonc 06 49 miles O 50 499 miles 500 999 miles 1000 3 161 miles o Pan o Allas Saturn Prometheus Pandora Epimethen Jamus Mimas Enceladus May have Waterie Alber Beth Tesly o Telest Calyp Dione o Helene Rhea Titan Hyperi Lapetus O Phoebe

Translate the following argument into symbolic form Then determine whether the argument is valid or invalid f you are happy then you smile If you smile then you do not go to the hockey game Therefore if you are happy then you go to the hocke Write the statement in symbolic form s the given argument valid or invalid O Invalid Valid SIDD

For the argument below perform the following a Translate the argument into symbolic form b Determine if the argument is valid or invalid Compare the argument to a standard form or use a truth table We will have tacos for dinner or we will have Chinese food We will not have tacos for dinner We will have Chinese food for dinner a Let p be We will have tacos for dinner and let q be We will have Chinese food for dinner Write the argument in symbolic form Choose the correct ans OA pvq OB pvq P P P q OC p q P P D is the argument valid or invalid A The argument is invalid because it is an example of Fallacy of the Converse B The argument is valid because it is an example of the Law of Syllogism C The argument is valid because it is an example of Disjunctive Syllogism OD The argument is invalid because it is an example of Fallacy of the Inverse OD p q p q

the argument below perform the following Translate the argument into symbolic form Use a truth table to determine whether the argument is valid or invalid Ignore differences in past present and future tense ve re lucky we ll win the contest we re not lucky We won t win the contest Let p be We re lucky and let q be We win the contest What is the argument in symbolic form A p q P q C p q P 9 the given argument valid or invalid C A The argument is valid because its corresponding statement is a tautology B The argument is valid because its corresponding statement is not a tautology C The argument is invalid because its corresponding statement is not a tautology D The argument is invalid because its corresponding statement is a tautology OB p q P 9 OD p q P 9

a The pants are not on sale but the necklace is on sale b If the pants are on sale then the necklace is on sale c It is false that the pants are on sale or the necklace is not o Choose the correct answer below O A Statements a and c are equivalent OB Statements a b and c are equivalent OC Statements b and c are equivalent O D Statements a and b are equivalent O E None of the statements are equivalent

I pay the bills but take the kids to the park or my wife is happy p I pay the bills q I take the kids to the park r My wife is happy Construct a truth table for the symbolic statement just found P T T T T F F F F 9 T T F F T T F F r T F T F T F T F Y Use the truth table to indicate the set of conditions that make the compound statement true or state that no such conditions exist OA The statement is true when p q and rare true B The statement is true when p q or r is true OC The statement is true when p and q are true or r is true OD The statement is true when p or q is true and r is true OF There are no conditions for which the statement in thin compound statement T Y

Write the following compound statement in terms of the simple statements p q and r Then construct a truth table for the compound statement is true I pay the bills but take the kids to the park or my wife is happy p I pay the bills q I take the kids to the park r My wife is happy Write the statement in symbolic form in terms of p q and r OA pvQ Ar OC pa Construct a truth table for the symbolic statement just found P T T q T T MUD OB PAG Vr OD pvajvr r T F 44 compound statement

All hawks fly Choose the correct answer below OA No hawks fly OB Some hawks do not fly OC Some hawks fly O D All hawks do not fly

Consider the following linear inequality 5z 17 13 6 4z Step 1 of 2 Solve the inequality and express your answer in interval notation Use decimal form for numerical values

Find the equation of the tangent line to the curve when x has the given value f x 3x 5x 7 x 2 O A y 7x 19 OB y 1 O c y 7x 28 X x 1 OD Y 2 y I 1 2

Previous Problem Problem List Next Problem 1 point Answer the following questions about the rooted tree shown below E K O PO Q B FO G L C 3 What is the level of H H M 2 What is the height of the graph 5 What is the parent of K For questions in which you need to list more than one vertex list the vertices in alphabetical order and separate them with a comma no spaces 1 Which vertex is the root 4 List the children of B If B has no children write NONE 6 What are the sibling s of E N 7 What are the ancestor s of Q I 8 What are the descendant s of C 9 List the leaves of the graph

3 Imbedding of Sobolev spaces with the same order of the derivative k and dif ferent integration powers Let 0 and p q 1 00 with q p Then it is Wka 2 CWk p S 3 points c Rd be a bounded domain k

ate the following argument into symbolic form Then determine whether the argument is valid or invalid sh are healthy but the water is not clear If the water is clear then the fish are healthy Therefore the wat the statement in symbolic form

slate the argument into symbolic form Then determine whether the argument is nows on Saturday then we will go to class It does not snow on Saturday Therefore we will not g be it snows on Saturday and let q be we will go to class What is the argument into symbolic fe

Please upload the proof to the following Theorem if h n g n and h n O g n and h n f n h n for all sufficiently large n then f n E g n

Simplify the following union and or intersection 8 8 17

What does the margin of error tell you in this situation A Gallup Poll in November 2014 found that 59 of the people in its sample said Yes when asked Would you like to lose weight Gallup announced For results based on the total sample of national adults one can say with 95 confidence that the margin of sampling error is 13 percentage points O That the actual percentage of people is 3 O That the person is really only 92 confident O That the actual percentage of people is between 56 and 62 That the people said they wanted to lose 3 of their body weight