Math 187 Sample Test II Questions

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1 Math 187 Sample Test II Questions Dr. Holmes October 2, 2008 These are sample questions of kinds which might appear on Test II. There is no guarantee that all questions on the test will look like these! This document is much longer than your test. I will distribute solutions to some of these later (maybe not all, as there is a quite a lot of duplication of kinds of question). There are some parts of questions which require ideas from section 17, which will not be on this test. 1. Venn diagrams (a) Illustrate the identity A (B C) = (A B) (A C) using Venn diagrams. Clearly label all sets in your diagrams, using keys to shadings or colors when appropriate. Clearly indicate which set in each diagram is the final result. (b) The equation A (B C) = (A B) (A C) is not an identity. Find finite sets A, B, C for which this equation is not true (and show by calculation that it is not true for these sets). You can use a Venn diagram to help find a counterexample, as we discussed in class. 2. A state has license plates consisting of four letters followed by three digits. 1

2 You need to set up calculations for your answers to this problem, but if you don t have a calculator (or if the number is too large for your calculator) the set-up by itself is enough. (a) If no additional conditions are imposed, how many license plates are possible? (b) How many license plates are possible if no letter can appear more than once and no digit can be followed immediately by the same digit? (c) How many license plates are possible if repetitions of letters and digits are allowed and at least one A or at least one 8 appears on the plate? Hint: think about plates which contain neither an A nor an Sets and set notation (a) Write the set {x Z 4 x x 16} in the notation which lists all its elements. (b) Write the set {x x {1, 2, 3} x 2} in the notation which lists all of its elements. Hint: this set has four elements, and the elements are sets, not numbers. 2

3 (c) Consider the two statements for all sets A, B, and C, if A B and B C, then A C and for all sets A, B, and C, if A B and B C, then A C. One of these statements is true: tell me which one (you do not need to prove it). Give examples of sets A, B, C which are a counterexample to the other statement. 4. License plates for a small state consist of three letters (A Z) followed by three digits (0 9). Show the setup for your calculations and give the final answer if you have a calculator. (a) How many license plates are possible if no other conditions are imposed? (b) How many license plates are possible if no letter can appear more than once on the plate and no digit can be immediately followed by the same digit? (c) Allowing unlimited repetitions, how many license plates contain at least one letter A? 5. Set theory notation: in each of the following sentences, write (is a member of) or (is a subset of) as appropriate. (a) 2 {1, 2, 3} (b) {1, 2, 3, 4} (c) {1, 2} {1, 2} (d) {1} {{1}} 6. (a) How many anagrams can be made from the word ABRACADABRA? Set this up in terms of factorials then compute it. 3

4 (b) A chain of beads is to be made with 12 beads, 4 beads of each of 3 different colors. How many different ways are there to do this? Set this up and compute it. 7. (a) A committee with 20 members is to choose a subcommittee with 5 members. How many ways are there to do this? (Set up and compute actual number). (b) A committee with 20 members is to choose a committee with 5 members which will have a chair and a secretary. How many ways are there to do this? (Set up and compute actual numbers). (c) A committee of 12 members is to divide itself into 3 working groups each with 4 members. How many ways are there to do this if the groups are labelled Rules, Membership, Finance? How many ways are there to do this if the groups have no special identifiers? Set up and compute actual numbers. 8. Write the first four terms of the expansion of (x+y) 20. Set this up using binomial coefficients, then evaluate the coefficients as actual numbers. 9. As is so often the case in my tests and examples, members of a group of 21 students are all enrolled in at least one of Math, French and English. 10 students take Math. 9 students take French. 13 students take English. 3 students take Math and French. 6 students take English and Math. 4 students take English and French. Either compute the number of students taking all three courses or prove that I am lying. Your solution must use the inclusion-exclusion principle: set up your work in such a way that I can clearly see that you used it. 4

5 Write the inclusion-exclusion formula for A B C D, where A, B, C, D are any four sets. 10. Consider the relation x F y on integers defined as 5 (x y). Write out the three things you need to prove to show that this is an equivalence relation (your answer must not depend on my knowing what words like reflexive, symmetric, and transitive mean: show me that you know what they mean). Prove one of them (if you prove more your best effort will count, and you might get a little extra credit). If the relation is restricted to the numbers between 1 and 20 inclusive, write out the equivalence classes under this relation (hint: numbers are equivalent if they have the same remainder on division by 5). 11. Give a demonstration of the theorem A (B C) = (A B) (A C) using Venn diagrams. Provide keys to your diagrams and clearly indicate the sets which are the final answers. 5

6 12. Write down the equivalence classes for the equivalence relation x R y defined as x y is divisible by 4 on the set of positive integers less than or equal to 20 (these are sets). 13. In how many different ways can the letters in REARRANGED be rearranged? Set up your answer in terms of factorials then compute an actual number. 14. You have a bin containing 12 beads of each of 4 different colors. You are going to make an open chain of just 12 beads (not 48). In each part set up your answer in terms of powers, factorials or binomial coefficients (as appropriate) then compute an actual number. (a) How many different chains of 12 beads with 3 of each color can you make? (b) How many handfuls of 12 beads of assorted colors are possible (here all that matters is how many beads you have of each color)? (c) How many chains of 12 beads of any mixture of colors are possible? 15. A committee of 10 people is appointing a subcommittee of 5 people. In each part, set up your calculation using binomial coefficients then compute an actual number. The last part is of course unrelated, but you still need to show your answer first using binomial coefficients then using actual numbers. (a) How many subcommittees of 5 are possible? 6

7 (b) Suppose we also need to specify a chair and a secretary from among the members of the subcommittee. How many possible ways are there to choose a 5 member subcommittee with chair and secretary? (c) Compute the first four terms of the expansion of (x + y) Of a group of 26 students, all must take English, Math, or French. 12 take English, 14 take Math, 14 take French. 7 take English and French, 3 take English and Math, 6 take French and Math. How many are taking all three subjects? You must set up and solve the problem in a way which demonstrates that you understand the Inclusion-Exclusion Principle. 17. Proofs. Do one of the following: if you do both, you will be graded on your best work. If you do very well on both you might get some extra credit. (a) Prove that the relation x R y on integers defined by x y is dvisible by 4 is an equivalence relation. There are three things to prove: set them up and identify each with an appropriate name, then prove them. (b) Prove using the appropriate proof templates from the book that for any sets A, B, C, if A B and B C then A C. Of course this is obvious, and you do not get noticeable credit for an informal explanation of the obvious; what is wanted is a formal proof in the style of the book or my board examples. 18. Venn diagrams (a) Give a pair of Venn diagrams illustrating the truth of A (B C) = (A B) (A C) Be sure to include a key to help me understand the shadings you use and to outline the final result sets clearly. 7

8 (b) Give a pair of Venn diagrams illustrating the fact that A (B C) = (A B) C is false, and give explicit examples of finite sets A, B, C for which it is false (your diagram can help you do this). 19. List all partitions of {1, 2, 3, 4} (a partition of a set A is a set of nonempty subsets of A such that each element of A belongs to exactly one element of the partition). 20. How many different anagrams can be made from the word ANAGRAMS? From the word DIFFERENT? 21. You want to make a necklace with 20 beads of 5 different colors. The necklace has a clasp past which beads can t be moved. How many ways can the necklace be made if you have 4 beads of each color? How does this change if there is no clasp, so beads can be moved freely all the way around the necklace? Suppose for the rest of the problem that you have 20 beads of each color (so you can use as many beads of each color as you like, but still no more than 20 beads in the necklace in all)? In how many ways can you choose a handful of 20 beads to put on the 8

9 necklace (this is just a handful, order doesn t matter). In how many ways can you make a necklace of 20 beads? 22. How many subcommittees with five members can be formed from a committee with ten members? How many ways can the subcommittees be chosen if each subcommittee has a chair and a treasurer chosen from among its five members? Write the first four terms of the expansion of (x + y) 1 0 (here I do want the coefficients fully computed, not just set up, and certainly not in binomial coefficient notation). 23. At a foreign language institute, 16 students study French, 19 study German, 17 study Russian, 8 study French and German, 7 study German and Russian, 9 study French and Russian. Of the 31 students, how many study all three languages? You are required to set this up and solve it using the inclusion-exclusion method, in a way which makes it clear that you understand this method. 24. Properties of relations (a) Consider the relation x is a full sibling of y on human beings (x and y are brothers or sisters, excluding half-brothers and halfsisters). Is this relation reflexive? irreflexive? symmetric? an- 9

10 tisymmetric? transitive? Explain each of your answers briefly. Some of these are mildly tricky. (b) Consider the relation x T y on integers defined by 3 (x y) (x y is divisible by 3, or x y mod 3). Prove that this is an equivalence relation (there are three things you need to prove: list them (clearly identifying them by name) and prove them). List the equivalence classes under this relation if we restrict our attention to positive integers less than

Chapter 3. Set Theory. 3.1 What is a Set?

Chapter 3. Set Theory. 3.1 What is a Set? Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or described. Any