Liberal Arts

1. Liberal Arts Mid-Term Review

2. Mid-Term Review • List the four steps in Polya’s problem-solving method. • Understand the problem • Devise a plan • Carry out your plan • Check your answer

3. Mid-Term Review • Draw a diagram that represents a rectangle whose length is 12 times the size of its width

4. Mid-Term Review • Consider the following diagram, which represents various groups of people. A is the group of people who like hot-dogs, and B is the group of people who like hamburgers. U represents everyone at the tail-gate. Describe the people in each region. r1 represents those who like hot-dogs and not hamburgers r2 represents those who like hamburgers and not hot-dogs r3 represents those who like both hd’s and hambgrs r4 represents those who like neither

5. Mid-Term Review • List all of the combinations of heads and tails if a nickel or penny are flipped.

6. Mid-Term Review • Complete the following sequences for the next 4 terms… • 3, 5, 9, 17, 33… • 1, 4, 9, 16, … • (a + 1), (b + 3), (c + 4)…

7. Mid-Term Review • Solve the following problem by making a series of guesses. In August of ’07, three of the top-selling video game players were the Wii, Xbox 360, and PS3. The total sales (in thousands) among the three were 812 units. The Xbox sold 15 more than twice the PS3, while the Wii sold 4 less than Xbox and PS3 combined. How many Wiis were sold.

8. Mid-Term Review • 405

9. Mid-Term Review • Explain the difference between inductive and deductive reasoning • Inductive reasoning is using patterns and observations to define an action • Deductive reasoning is using proven facts, laws, and truths to define an action

10. Mid-Term Review • Use an alternative method to write each set: • T = {A-, A+, B-, B+, AB-, AB+, O-, O+} • B = {color of the American flag} • A = {a: a is a counting number less than 20 and is evenly divisible by 3}

11. Mid-Term Review • Determine if the following are well-defined or not well-defined sets. • {x: x lives in Michigan} • Well defined • {2, 4, 6, 8…} • Well defined • {t: t has traveled much} • Not well defined • {y: y has an easy cell phone number to remember} • Not well defined

12. Mid-Term Review • Give an example of the following: • Give a statement such that the answer has an empty set { } • T = {x: x are students @ LLT that were born before 1980} • Define Universal set: • The set of all elements under consideration in a given discussion; U

13. Mid-Term Review • Find the cardinal number, n(A)=…, for the following sets • A = {1, 3, 5, 7… 11} • n(A) = 6 • A = {3, 4, 5… 13} • n(A) = 11 • A = {x: x is a living American president born before 1900} • n(A) = 0

14. Mid-Term Review • Describe each of the following sets as finite or infinite: • {x: x is a word written by Shakespeare} • Finite • {y: y is the number of people who have walked on the moon} • Finite • {y: y is a number between 4 & 5} • Infinite • {x: x is an element of the empty set} • Finite

15. Mid-Term Review • Explain the difference between subset and proper subset. • Subsets are when all elements of A are also in B; however, they do not have to equal • Proper subsets are when all elements of A are also in B, but they cannot be equal

16. Mid-Term Review T = set of 100-m sprinters A = set of U.S. athletes G = set of gold medal winners True or False?

17. Mid-Term Review T = set of 100-m sprinters A = set of U.S. athletes G = set of gold medal winners True or False?

18. Mid-Term Review T = set of 100-m sprinters A = set of U.S. athletes G = set of gold medal winners True or False?

19. Mid-Term Review T = set of 100-m sprinters A = set of U.S. athletes G = set of gold medal winners Using the definitions of subsets and proper subsets, give an example of a true statement

20. Mid-Term Review • Determine if the following sets are equal and equivalent: • {1, 2, 3, 4} and {4, 3, 2, 2} • Not equal; equivalent • {a, b, c…m} and {1, 2, 3…13} • Not equal; equivalent • {1, 3, 5, 2, 4, 6} and {1, 2…6} • Equal; equivalent • {red, white, blue} and {colors} • Not equal; not equivalent

21. Mid-Term Review • What is another name for the term equivalent? • One-to-one correspondence

22. Mid-Term Review • Determine whether each statement is true of false: • {collie, poodle, beagle, chihuahua, bulldog} is a proper subset of {bulldog, chihuahua, beagle, poodle, collie} • False • {rose, daisy, sunflower, orchid} is a proper subset of {rose, orchid, sunflower, daisy, carnation} • True

23. Mid-Term Review • List all of the two-element subsets of the set {1,2,3,4} • {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} • List all of the subsets for of the set {1,2,3,4} • { }, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {2,3,4}, {1,3,4} {1,2,3,4}

24. Mid-Term Review • Domino’s Pizza wants to advertise that there are over 500 ways to order your pizza, how many different toppings must they have available? • At least 9 toppings

25. Mid-Term Review • If a set has 30 elements, there will be 2^30 = 1,073,741,824 subsets. If you could write one subset every second, how long would it take you to list all 2^30 subsets? (Hint: There are 365 x 24 x 60 x 60 seconds in a year – ignoring leap years) • Approximately 34.9 years

26. Mid-Term Review • Draw each of the following set operations:

27. Mid-Term Review • Find the intersection of the two sets.

28. Mid-Term Review • Given U, find the complement of A.

29. Mid-Term Review • Let U = {1,2,3,4,5}, A = {1,3,5}, B = {1,2,3} • Find (A U B)’ • {4} • Find A’ U B’ • { }

30. Mid-Term Review • Let U = {1,2,3,… 10}, E = {x: x is even}, B = {1,3,4,5,8}, and A = {1,2,4,7,8} • Find (A U B)’ intersect (E’ U A)

31. Mid-Term Review • Example: A television network conducted a market survey. The following information on viewing habits was obtained. Determine how many people do not want to see reality shows and how many prefer to watch TV on the weekend?

32. Mid-Term Review • The universal set contains three subsets.

33. Mid-Term Review • Do not want to see reality shows = 12 + 11 + 4 + 13 = 40 • Prefer to watch TV on weekend =14 + 18 + 4 + 13 = 49