( ) = A. 2. Write the following sets using the roster method. 3. Write the following sets using set-builder notation.

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1 2.6. EXERISES 1. True or False? a. The empty set has no subsets. b. No set has exactly 14 distinct subsets. c. For any two finite sets and, <. d. If and are both proper subsets of, then the union of and is also a proper subset of. e. The set belongs to the power set of. ( ) =. f. For any two sets and, g. If, then =. h. If =, then. i. If, then. 2. Write the following sets using the roster method. a. The set of letters in the word mathematics. b. The set of natural numbers with two digits that are multiples of 13. c. The set of pentagons with six sides. d. n n W and n <12 { } { } e. 2n +1 n N and n < Write the following sets using set-builder notation. a. The set of natural numbers with exactly two digits b. The set of natural numbers divisible by 7. c. The set of real numbers larger than, or equal to, d. { 5,25,125,625,... } e. 5,15,25, 35, 45,55,... { } 4. Specify which of the following sets are well defined. a. The set of New York State senators currently under investigation for an alleged crime. b. The set of small countries in the European nion. c. The set of real numbers not equal to their cube. d. The set of college students on Long Island who are excellent cooks. e. The set of all sets (also know as the universal set ξ).

2 5. Find the cardinality of the following sets. a. The set of face cards in a standard deck of cards. b. The set of positive integers with at most 2 digits. c. The set of positive integers divisible by 10 with exactly 3 digits. d. { 5,25,45,65,85,...,205} e. The set of.s. states that border the tlantic Ocean and Kentucky. 6. Find the cardinality of the power set for each of the sets in Exercise How many different pizzas with at least two toppings can you create from a selection of seven toppings? ssume you can use any, all, or none of the seven toppings. 8. Your friend just got back from a set theory lecture and wants to impress you. She claims that the number of distinct possibilities you have of grabbing any number of cards randomly out of a standard deck of 52 cards (so you could grab no cards, or one card, or two cards, or any other number of cards up to the entire deck) is larger than a quadrillion, or Is she actually correct? For reference, this order of magnitude, which is equal to a million billions, roughly corresponds to the number of synapses in a human brain or the number of ants walking the planet at any given time. 9. Explain why = for any set. Illustrate this property with an example. 10. Explain why the absorption law, which states that ( ) =, must be valid for any sets and. Illustrate this property with an example. 11. Let N be the set of letters in the word Nassau and S be the set of letters in the word Suffolk. Find the following: a. N S b. N S c. P N S d. P N S { } and 12. Suppose is a set that consists of odd integers. Find if 1,5 { 3, 1,7 } are both proper subsets of and has exactly 16 distinct subsets. Explain your reasoning.

3 13. Let the universal set be the set of letters in the word uncopyrightable (the longest word in the English language with no repeating letters). Let be the set of vowels in (assume that y is not a vowel). Let = { a,c,o, p,t,u,y}. Find the following: a. b. c. { } and 14. Suppose is a set that consists of odd integers. Find if 1,5 { 3, 1,7 } are both proper subsets of and has exactly 16 distinct subsets. Explain your reasoning. 15. Let be the set of natural numbers between 1 and 20, including both 1 and 20. Suppose is the set consisting of all even numbers in, is the set consisting of all prime numbers in, and = { 1,3,6,8,11,15,16,19}. Find the following sets: a. b. ( ) c. ( ) 16. Suppose S = { 1,2,3,...,11,12} and D is a set containing 10 natural numbers. Find a set D that satisfies the following conditions: a. S D = 22 b. S D =17 c. S D =12 d. S D =10 e. S D = 5 f. S D = The following is the Venn diagram for two sets and when. Find the three pairs of equal sets in the following list:,,,,,

4 18. If two sets and are disjoint, find a formula for + in terms of and/or. 19. If is a subset of (see Venn diagram in Exercise 7), find a formula for each of the following in terms of and/or : a. b. c. 20. Prove the set identity ( ) =. se the indexing of the Venn diagram given below and shade the regions that correspond to both sides of the identity Prove De Morgan s first law, which states that =. se the indexing of the Venn diagram given below and shade the regions that correspond to both sides of the identity

5 22. Prove the set identity ( ) = ( ). se the indexing of the Venn diagram given below and shade the regions that correspond to both sets Prove that ( ) ( ). se the indexing of the Venn diagrams given below and shade the regions that correspond to each side of the property

6 24. a. Write a set expression that corresponds to the shaded regions in the Venn diagram above. b. Write an equivalent set expression that uses the four basic set operations of complementation, union, intersection, and difference. 25. Write a set expression that corresponds to the shaded regions in the following Venn diagrams. a.

7 b. c. d.

8 Survey Problems 1. recent survey of 120 Long Island restaurants found that 67 of them feature Italian dishes, 35 of them feature French dishes and 32 of them feature neither Italian nor French dishes. a. Summarize the results of this survey in the Venn diagram below. Label your sets. b. How many restaurants feature both Italian and French dishes? c. How many restaurants do not feature any French dishes? 2. Suppose three brothers have five siblings each. How many children does this family have in total? Justify your answer using a Venn diagram with three sets. 3. The Financial id Director of a small community college surveyed the records of all students and came up with the following data: 49% of the students receive private aid. 43% of the students receive scholarships from the college. 23% of the students receive both government grants and private aid. 18% of the students receive both government grants and scholarships from the college. 28% of the students receive both private aid and scholarships from the college. 8% of the students receive help from all three sources.

9 Summarize the results of this survey in the Venn diagram given below and answer the given questions. a. What percentage of students receive government grants only? b. What percentage of students receive private aid but not government grants? c. What percentage of students receive financial aid from only one of the three sources? d. What percentage of students receive no aid from the college or from the government? e. What percentage of students receive no financial aid from any of these sources? Government ollege P rivate 4. In a logic class, 13 students are teenagers, 17 students own a car, and 9 have traveled abroad. If exactly 7 students are over 19 years old, do not own a car and have traveled outside the nited States, then how many students are teenagers who own a car?

10 Mind ogglers 1. Is the set of all sets that can be described in sixty letters or less well defined? 2. single set partitions the universe into two distinct regions: and. Two sets partition the universe into at most four distinct regions. Three sets partition the universe into at most eight distinct regions. a. Explain how three sets could partition the universe into exactly six distinct regions. Draw a Venn diagram that corresponds to this special case. b. What is the maximal number of regions that would need to be indexed if n sets partition the universe? 3. Suppose is a set with 45 elements, is a set with 27 elements, and is a set with 20 elements. a. What is the maximum possible number of elements in? How are the three sets related in this case? Illustrate this with an example. b. What is the minimum possible number of elements in? How are the three sets related in this case? Illustrate this with an example. c. What is the maximum possible number of elements in ( )? How are the three sets related in this case? Illustrate this with an example. 4. Here s a neat question formulated by the logician Raymond M. Smullyan. Suppose you and I are immortal. I write down a natural number on a piece of paper and grant you one (and only one) chance each day to guess the number. If you manage to guess the number, I offer you free backrubs for the rest of time. a. What strategy would allow you to guess this number eventually? b. Suppose I now write down either a positive or a negative integer on the piece of paper. nswer the same question now. c. Suppose I now write down two natural numbers (they could be identical) on the piece of paper. What strategy would allow you to guess both numbers eventually on a given day?