MGF 1106Test 1 ReviewandPractice Solutions

• You will need to have your own calculator for the test. • You may not share calculators or use any type of communication device in place of a calculator. • Tests may not be made up for any reason other than a mandatory school – sponsored activity for which you must miss class. • If you miss one test for any other reason, your final exam score will be substituted for that test. A second missed test is a zero. No homework bonuses are awarded on a test when the final exam is substituted or you receive a zero on a missed test. • It is always possible that some questions may not be exactly like those on the practice test (objectives will remain unchanged). Some of these questions may ask you to analyze or synthesize what we’ve learned but if you prepare by studying your notes and your worksheets (as well as this review), you will be ready to complete those questions as well.• Be sure you know the vocabulary listed in the Test 1Vocabulary Handout. Some vocabulary questions will be on the exam.

Questions on the test will be in a random order and not in the order that appears in this review.

Be sure to complete each assignment with a score of at least 80% to receive the 10 point bonus.

Exam TopicsObjective Section Suggested Text Problems1) Use inductive reasoning to find the next number in a sequence. 1.1 Page 43: 3, 5, 7, 9

2) Use deductive reasoning. 1.1 Page 43: 14Answer to b: 3) Round numbers. 1.2 Page 43: 154) Work cost comparison problems. 1.3 Page 39: 355) Work coin combination problems 1.3 Page 39: 28 (answer: 12)6) Work problems involving runners in a race. 1.3 Page 39: 397) Use the symbols ∈ and ∉. 2.1 Page 109: 98) Find the cardinality of a set. 2.1 Page 109: 119) Answer True/False Questions about subsets. 2.2 Page 68: 5110) List all the subsets of a given set. 2.2 Page 68: 55, 5711) Find the number of subsets and proper subsets of a set. 2.2 Page 69: 9312) Find the complement of a set. 2.3 Page 81: 5, 713) Perform operations on two sets. 2.3 Page 109: 33, 35, 3714) Use a Venn diagram to answer questions about a set. 2.3 Page 82: 109, 11115) Place items in a Venn diagram. 2.4 Page 94: 99, 101, 10316) Work survey problems. 2.5 Page 110: 6117) Use a Venn diagram to answer questions about a survey. 2.5 Page 104: 33, 35, 3718) Negate quantified statements. 3.1 Page 207: 13, 1519) Translate a compound statement to symbolic form. 3.2 Page 207: 7, 920) Translate symbolic form to a compound statement. 3.2 Page 207: 3

Important Ideas1) Use inductive reasoning to find the next number in a sequence. Look for a pattern. Choose the number that best fits the pattern.2) Use deductive reasoning. Find the indicated numbers.Deduce a pattern.

Prove your deduction using algebra.3) Round numbers. If the digit to the right is 5, round up. Otherwise, stay the same.4) Work cost comparison problems. Calculate the cost for each situation. Subtract to compare.5) Work coin combination problems Use an organized chart.6) Work problems involving runners in a race. Use a diagram.7) Use the symbols ∈ and ∉. ∈ - is an element of ∉ - is not an element of8) Find the cardinality of a set. The cardinality of set A is the number of elements of set A.9) Answer True/False Questions about subsets. ⊆ - subset of ⊂ - proper subset of (subset but not equal to)10) List all the subsets of a given set. Be sure to list the empty set, the sets of cardinality 1, 2, 3, etc., and the set itself11) Find the number of subsets and proper subsets of a set. If n represents the number of items in the set, the number of subsets is and the number of proper subsets is 12) Find the complement of a set. A’ = {x| x ∈ U and x ∉A}13) Perform operations on two sets. A⋂B = {x| x∈A and x∈B} A⋃B = {x| x∈A or x∈B}14) Use a Venn diagram to answer questions about a set. Be sure to understand what the different areas of a Venn diagram represent.15) Place items in a Venn diagram. See above.16) Work survey problems. Within the Venn diagram, work inside – out. Check your work.17) Use a Venn diagram to answer questions about a survey. The numbers in each region correspond to the cardinality of each

region – how many elements are in the region.18) Negate quantified statements. All A are B. No A are B. Some A are B. Some A are not B.19) Translate a compound statement to symbolic form. ∧ And∨ Or → If then↔ If and Only If20) Translate symbolic form to a compound statement. See above.

Practice Test Solutions

1) Identify a pattern in the list of numbers. Describe the pattern. Then use this pattern to find the next number.

a) 7, 10, 13, 16, ?The difference between each number is 3.

b) 7, 14, 28, 56, ?The number doubles each time.

c) 2, 6, 12, 36, 72, 216, ?The numbers alternate between being multiplied by 3 and multiplied by 2.

2) Select a number. Multiply it by 6. Subtract 8 from the product. Divide the difference by 2. Add four to the quotient. a) Repeat this process for the numbers 1, 2, 5, 10.

b) Write a conjecture that relates the result of this process to the original number selected. The number is multiplied by 3 (tripled).c) Let represent the original number. Use deductive reasoning to prove your conjecture.

3) Round 7.899157 to the nearest:a) tenths: 7.9 b) thousandths: c) hundred – thousandths: 7.89916

4) John has a vending route. He’s paid $450 per week to service the vending machines. Kent’s company pays him $1750 per month to do the same job. In a single year, who makes more money? What is the difference between the two annual salaries? Assume 12 months in a year and 52 weeks in a year.John:Kent:

John makes more money. His annual salary exceeds Kent’s by $2400.

5) If you spend $9.74, in how many ways can you receive change from a ten-dollar bill. Use a table and show all possibilities.Quarters Dimes Nickels Pennies1 0 0 10 2 1 10 2 0 60 1 3 10 1 2 60 1 1 110 1 0 160 0 5 10 0 4 60 0 3 110 0 2 160 0 1 210 0 0 26

$

13 ways

6) Trent, Riley, Susan, Doug, and Betty competed in a race. Betty finished 11 seconds after Doug. Betty finished 24 seconds after Riley. Susan finished 8 seconds after Riley. Doug finished 18 seconds before Trent. In which order did the runners finish the race? How many seconds difference was there between first and last place? (First is on the right) 11 B D  11 13 B D R 11 5 8 B D S R    7 11 5 8 T B D S R   Order: Riley, Susan, Doug, Betty, Trent.31 seconds between first and last.

 7) Use the symbols ∈ and ∉ to make the statement true.a) 3 _____ {3, 4, 5, 6}3 __ ___ {3, 4, 5, 6}b) {7} ______ {}{7} __ ∉ ____ {}c) 9 ______ {} 9 __ ∉ ____ {}

8) Find the cardinal number for each set.a) b)

4

9) Determine if each statement is true or false.a) {1, 2} ⊆ {0, 1, 2, 3, 4, 5}True{1, 2} is a subset of {0, 1, 2, 3, 4, 5}

b) {1, 2, 3} ⊂ {}Falsec) { } ⊂ {1, 2, 3}TrueThe empty set is a proper subset of every set other than itself.

d) 1 ⊈ {1, 2, 3, 4}True1 is an element, not a subset.

10) List the subsets of {cake, pie, cookie}No elements (the empty set): { }

One element: {cake}, {pie}, {cookie}Two elements: {cake, pie}, {cake, cookie}, {pie, cookie}

Three elements (the set itself): {cake, pie, cookie}

11) Suppose S is the set How many subsets does S have? How many proper subsets does S have?

12) Suppose Find A’.Find B’.

13) Suppose .Find the following.a)

b) c) d)

14) Jacob was preparing to pack for college. He called his future roommate Marcus to ask if he had purchased certain items. He put the outcome of that phone conversation in the Venn diagram below. What is the set of items that Jacob purchased but not Marcus?

{𝑡𝑜𝑎𝑠𝑡𝑒𝑟 , 𝑓𝑢𝑡𝑜𝑛 , 𝑡𝑒𝑛𝑡 }

15) Suppose A represents the set of letters in the word “canned”. Set B represents the set of letters in the word “processed”. Set C represents the set of letters in the word “organic”. In what region would you place the letter “e” ?

The letter “e” is in sets A and B but not in set C.The region that contains A and B but not C is II.

16) A survey of 180 high school students was taken to determine participation in various college activities. 43 were in a service club. 52 were in the band. 35 were on an athletic team. 13 were in a service club and the band. 14 were in the band and on an athletic team. 12 were in a service club and an athletic team. Five students participated in all three activities.

17) Pete’s Pet Shop surveyed dog owners to find out if they gave their pets canned dog food or pet treats. The results of the survey are summarized in the Venn diagram below.How many dog owners gave their pets canned dog food and pet treats?

The answer is the intersection of the two sets: 7.

18) Negate each statement.a) All whales are mammals. Some whales are not mammals. b) No blue clouds are in the sky. Some blue clouds are in the sky. c) Some pianists are not proficient.All pianists are proficient. d) Some children like video games.No children like video games.

19) Let p, q, and r represent the following statements. p: Jim likes oranges q: Mike likes strawberries r: Katie likes mangoesTranslate each statement to symbolic form.a) Jim likes oranges and Katie does not like mangoes. p ∧ ~r b) Katie likes mangoes if and only if Mike likes strawberries. r ↔ qc) If Jim does not like oranges, then Mike likes strawberries or Katie does not like mangoes. ~p→(q ∨ ~r)

20) Let p, q, and r represent the statements in question 19.p: Jim likes oranges q: Mike likes strawberries r: Katie likes mangoesTranslate each symbolic form to English.a) p ∨ q Jim likes oranges or Mike likes strawberries. b) q ∧ (~p →~r) Mike likes strawberries, and if Jim does not like oranges then Katie does not like mangoes.

MGF 1106Extra Practice Problems

1) Identify a pattern in the list of numbers. Describe the pattern. Then use this pattern to find the next number.

1) Identify a pattern in the list of numbers. Describe the pattern. Then use this pattern to find the next number.

2) Select a number. Multiply it by 5. Subtract 15 from the product. Divide the difference by 5. Add three to the quotient. a) Repeat this process for the numbers 1, 2, 5, 10.

b) Write a conjecture that relates the result of this process to the original number selected. c) Let represent the original number. Use deductive reasoning to prove your conjecture.

2) Select a number. Multiply it by 5. Subtract 15 from the product. Divide the difference by 5. Add three to the quotient. a) Repeat this process for the numbers 1, 2, 5, 10.

b) Write a conjecture that relates the result of this process to the original number selected. The result is the original number.c) Let represent the original number. Use deductive reasoning to prove your conjecture.

3) Round 4.95834 to the nearest hundredth.

3) Round 4.95834 to the nearest hundredth.4.96

4) Rayna works as a math tutor and earns $2,495 each month. Her sister Jeanine gives private piano lessons and charges $45 per hour. If Jeanine gave piano lessons for 57 hours last month, who made more money? How much more?

4) Rayna works as a math tutor and earns $2,495 each month. Her sister Jeanine gives private piano lessons and charges $45 per hour. If Jeanine gave piano lessons for 57 hours last month, who made more money? How much more?Rayna:Jeanine

Jeanine made more money. She earned 70 more dollars.

5) If you spend $9.68, in how many ways can you receive change from a ten-dollar bill? Use a table and show all possibilities.

5) If you spend $9.68, in how many ways can you receive change from a ten-dollar bill? Use a table and show all possibilities.$

18 ways

Quarters Dimes Nickels Pennies1 0 1 21 0 0 70 3 0 20 2 2 20 2 1 70 2 0 120 1 4 20 1 3 70 1 2 120 1 1 170 1 0 220 0 6 20 0 5 70 0 4 120 0 3 170 0 2 220 0 1 270 0 0 32

6) Four mascots competed in a race at a baseball game. The dancing bear finished eight seconds ahead of the chicken. The chicken finished 3 seconds behind the penguin. The ostrich finished 12 seconds ahead of the penguin. Who finished first? What was the difference between first place and last?

6) Four mascots competed in a race at a baseball game. The dancing bear finished eight seconds ahead of the chicken. The chicken finished 3 seconds behind the penguin. The ostrich finished 12 seconds ahead of the penguin. Who finished first? What was the difference between first place and last? (First is on the right) 8 C B  3 5 C P B 3 5 7 C P B O     Order: Ostrich, Bear, Penguin, Chicken15 seconds between first and last.

 7) Use the symbols ∈ and ∉ to make the statement true. c _____ {}

 7) Use the symbols ∈ and ∉ to make the statement true. c ___ ∈ _ {}

8) Find the cardinal number for the set.{}

8) Find the cardinal number for the set.{}

9) Determine if the statement is true or false. ⊂{}

9) Determine if the statement is true or false. ⊂{}True

10) List the subsets of {tea, milk, water, soda}

10) List the subsets of {tea, milk, water, soda}No elements (the empty set): { }

One element: {tea}, {milk}, {water}, {soda}Two elements: {tea, milk}, {tea, water}, {tea, soda}, {milk, water}, {milk, soda}, {water, soda}

Three elements: {tea, milk, water}, {tea, milk, soda}, {tea, water, soda}, {milk, water, soda}Four elements (the set itself):{tea, milk, water, soda}

11a) How many subsets of {tea, milk, water, soda} should you have found? Proper subsets?

11a) How many subsets of {tea, milk, water, soda} should you have found? Proper subsets?

11b) Jacob is going on vacation. He is thinking about bringing some of his textbooks on the trip. He has five textbooks and will bring some, all, or none of the textbooks. How many different options does he have?

11b) Jacob is going on vacation. He is thinking about bringing some of his textbooks on the trip. He has five textbooks and will bring some, all, or none of the textbooks. How many different options does he have?

The number of options is the number of subsets of the set of his five textbooks.

12) Find A’:

12) Find A’:

𝐴′={𝑝𝑙𝑎𝑛𝑒 , 𝑓𝑒𝑒𝑡 ,𝑏𝑎𝑙𝑙𝑜𝑜𝑛}

13) Find ,

13) Find ,

14) Jacob was preparing to pack for college. He called his future roommate Marcus to ask if he had purchased certain items. He put the outcome of that phone conversation in the Venn diagram below. What is the set of items they both purchased?

14) Jacob was preparing to pack for college. He called his future roommate Marcus to ask if he had purchased certain items. He put the outcome of that phone conversation in the Venn diagram below. What is the set of items they both purchased?

{𝑔𝑢𝑖𝑡𝑎𝑟 ,𝑚𝑖𝑐𝑟𝑜𝑤𝑎𝑣𝑒 ,𝑡𝑒𝑙𝑒𝑣𝑖𝑠𝑖𝑜𝑛 }

15) Suppose A represents the set of letters in the word “canned”. Set B represents the set of letters in the word “processed”. Set C represents the set of letters in the word “organic”. In what region would you place the letter “g” ?

15) Suppose A represents the set of letters in the word “canned”. Set B represents the set of letters in the word “processed”. Set C represents the set of letters in the word “organic”. In what region would you place the letter “g” ?

The letter “g” is in set C but not in A or B.The region that contains C but not A or B is VI.

16) 100 people were surveyed about what sport they like they like. 35 people like basketball. 31 people like baseball. 59 people like football. 16 people like basketball and baseball. 25 like basketball and football. 20 like baseball and football. 14 like all three sports. How many like exactly one sport?

16) 100 people were surveyed about what sport they like they like. 35 people like basketball. 31 people like baseball. 59 people like football. 16 people like basketball and baseball. 25 like basketball and football. 20 like baseball and football. 14 like all three sports. How many like exactly one sport?

45 like exactly one sport.

17) Pete’s Pet Shop surveyed dog owners to find out if they gave their pets canned dog food or pet treats. The results of the survey are summarized in the Venn diagram below.How many dog owners were surveyed?

17) Pete’s Pet Shop surveyed dog owners to find out if they gave their pets canned dog food or pet treats. The results of the survey are summarized in the Venn diagram below.How many dog owners were surveyed?

Answer:

18) Negate each statement.a) All squirrels can climb.  b) Some people like the NFL.  c) Some tests are not graded on a curve. d) No roller coasters are scary.

18) Negate each statement.a) All squirrels can climb. Some squirrels cannot climb. b) Some people like the NFL. No people like the NFL. c) Some tests are not graded on a curve.All tests are graded on a curve. d) No roller coasters are scary.Some roller coasters are scary.

19) Let p, q, and r represent the following statements. p: Jim likes oranges q: Mike likes strawberries r: Katie likes mangoesTranslate the statement to symbolic form.If Mike doesn’t like strawberries or Jim doesn’t like oranges, then Katie doesn’t like mangoes and Jim likes oranges.

19) Let p, q, and r represent the following statements. p: Jim likes oranges q: Mike likes strawberries r: Katie likes mangoesTranslate the statement to symbolic form.If Mike doesn’t like strawberries or Jim doesn’t like oranges, then Katie doesn’t like mangoes and Jim likes oranges.(~q ∨ ~p )→(~r ∧ p)

20) Let p, q, and r represent the statements in question 19.p: Jim likes oranges q: Mike likes strawberries r: Katie likes mangoesTranslate the symbolic form to English.  (p ∧ q) ↔ ~r

20) Let p, q, and r represent the statements in question 19.p: Jim likes oranges q: Mike likes strawberries r: Katie likes mangoesTranslate the symbolic form to English.  (p ∧ q) ↔ ~r

Jim likes oranges and Mike likes strawberries, if and only if Katie does not like mangoes.