HW4 sol.pdf - 1.2 Let S ={\u22122 \u22121 0 1 2 3 Describe each of the following sets as{x \u2208 S p(x where p(x is some condition on x(a(b(c(d A ={1 2 3 B ={0

# HW4 sol.pdf - 1.2 Let S ={u22122 u22121 0 1 2 3 Describe...

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Unformatted text preview: 1.2. Let S = {−2, −1, 0, 1, 2, 3}. Describe each of the following sets as {x ∈ S : p(x)}, where p(x) is some condition on x. (a) (b) (c) (d) A = {1, 2, 3} B = {0, 1, 2, 3} C = {−2, −1} D = {−2, 2, 3} MATH 2345 section 54/55 Homework 4 Solutions 1.3. Determine the cardinality of each of the following sets: (a) A = {1, 2, Q1. Let = #1, {13,},4, ({5} 1})*. Write × . How many elements does ( × ) have? (Don’t write (b) B = {0, 2, 4, . . . , 20} it down!) (c) C = {25, 26, 27, . . . , 75} (d) D = {{1, 2}, {1, 2, 3, 4}} ×= (e) E = {∅} ((1,1), (1, {1}), /1, ({1})0, ({1} ,1), ({1}, {1}), /{1}, ({1})0, /({1}), 10, /({1}), {1}0, /({1}), ({1})0) (f) F = {2, {2, 3, 4}} () = #∅, {1}, ({1}), #({1})* , (1, {1}), #1, ({1})* , #{1}, ({1})* , #1, {1}, ({1})* * 1.4. Write each of the following sets by listing its elements within braces. 5 |( (a) × )| A =={n2∈ Z : −4 < n ≤ 4} 5} (b) B = {n ∈ Z : n2 < < < < < 3 Q2. Let = {n #∈∈Nℝ| ≤ ≤ 7 * = >− 7 , 7 ? for all positive integers . < (c) C7 = : n− 7 100} (d) D = {x ∈ R : x 2 − x = 0} 0}7 E =⋃ {x<B∈ R x 2 +⋂1<B= a)(e)Find 7: and 7C< 7C< ⋃<B 7C< 7 = [−1,1] < < ⋂<B 7C< 7 = [− <B , <B] 1.5. Write each of the following sets in the form {x ∈ Z : p(x)}, where p(x) is a property concerning x. G G b)(a)Find 7 and A =⋃ {−1, −3,⋂ . .7C< .} 7 7C<−2, ⋃G 7C< 7 = [−1,1] ⋂G 7C< 7 = {0} (b) B = {−3, −2, . . . , 3} c)(c)Are J , 1, … 2} mutually disjoint? No because ⋂G < , I ,−1, 7C< 7 = {0} C ={−2, 1.6. The set E = {2x : x ∈ Z} can be described by listing its elements, namely E = {. . . , −4, −2, 0, 2, 4, . . .}. List the elements of the following sets in a similar manner. Q3. A group of college students were asked about their TV watching habits. Of those surveyed, (a) A =watch {2x + The 1 : xWalking ∈ Z} 28 students Dead, 19watch The Blacklist, and 24 watch Game of Thrones. n ∈ Z}The Walking Dead and The Blacklist, 14 watch The Walking Dead and (b) B = {4n Additionally, 16 :watch = {3q + and 1 : q10 ∈ Z} Game(c)ofCThrones, watch The Blacklist and Game of Thrones. There are 8 students who watch allset three How surveyed at leastbyone of the 1.7. The E =shows. {. . . , −4, −2,many 0, 2, 4,students . . .} of even integerswatched can be described means of a shows? defining condition by E = {y = 2x : x ∈ Z} = {2x : x ∈ Z}. Describe the following sets in a similar manner. | ∪ ∪ | (a) A = {. . . , −4, −1, 2, 5, 8, . . .} = || + || + || − | ∩ | − | ∩ | − | ∩ | + | ∩ ∩ | (b) B = {. . . , −10, −5, 0, 5, 10, . . .} = 28 + 19 + 24 − 16 − 14 − 10 + 8 = 39 (c) C = {1, 8, 27, 64, 125, . . .} 1.8. Let A = {n ∈ Z : 2 ≤ |n|√< 4}, B = Q4. √ {x ∈ Q : 2 < x ≤ 4}, C = {x ∈ R : x 2 − (2 + (a) (b) (c) (d) (e) 2)x + 2 2 = 0} and D = {x ∈ Q : x 2 − (2 + √ √ 2)x + 2 2 = 0}. Describe the set A by listing its elements. Give an example of three elements that belong to B but do not belong to A. Describe the set C by listing its elements. Describe the set D in another manner. Determine the cardinality of each of the sets A, C and D. (c) A ∈ B and A ⊂ C. (c) Describe the set C by listing its elements. 1.11. Let (a, b) be an open interval of real numbers and let c ∈ (a, b). Describe an open interval I centered at c such I ⊆ (a, (d) that Describe theb). set D in another manner. 1.12. Which of the following sets are equal? (e) Determine the cardinality of each of the sets A, C and D. A = {n ∈ Z : |n | < 2} D = {n ∈ Z : n 2 ≤ 1} B = {n ∈ Z : n 3 = n } E = {−1, 0, 1}. Solution. C = {n ∈ Z : n 2 ≤ n } (a) The set A listed explicitly is A = {−3, −2, 2, 3}. 1.13. For a universal set U = {1, 2, . . . , 8} and two sets A = {1, 3, 4, 7} and B = {4, 5, 8}, draw a Venn diagram (b) represents Since the these elements that sets.in B are rational numbers, any three rational numbers (that are not simultaneously integers) in B will not be in A. Three such 1.14. Find P(A) and |P(A)| for examples are: 52 , 27 and 13 4 . (a) A = {1, 2}. √ √ √ (c) ANote that x2 − (2 + 2)x +√ 2 2 = 0 can be written as (x − 2)(x − 2) = 0. (b) = {∅, 1, {a}}. This implies that C = {2, 2}. 1.15. Find P(A) for A = {0, {0}}. (d) Since D is restricted to Q, we have D = {2}. 1.16. Find P(P({1})) and its cardinality. (e) P(A) The cardinalities are:A |A| = 4, ∅, |C| {∅}}.= 2, and |D| = 1. 1.17. Find and |P(A)| for = {0, 1.18. For A = {x : x = 0 or x ∈ P({0})}, determine P(A). 1.19. Remark. Give an example set S such that There of area infinite possibilities for part (b). For part (c), we could have optionally used the quadratic formula if the factorization is not readily (a) S ⊆ P(N) recognized. As for set D, note that Z ⊂ Q, so it is not necessary to write 2 as (b) S ∈ P(N) a fraction. (c) S ⊆ P(N) and |S | = 5 (d) S ∈ P(N) and |S | = 5 1.20. Q5. Determine whether the following statements are true or false. (a) (b) (c) (d) If {1} ∈ P(A), then 1 ∈ A but {1} ∈ / A. If A, B and C are sets such that A ⊂ P(B) ⊂ C and |A| = 2, then |C| can be 5 but |C| cannot be 4. If a set B has one more element than a set A, then P(B) has at least two more elements than P(A). If four sets A, B, C and D are subsets of {1, 2, 3} such that |A| = |B| = |C| = |D| = 2, then at least two of these sets are equal. 1.21. Three subsets A, B and C of {1, 2, 3, 4, 5} have the same cardinality. Furthermore, 1 belongs to A andB=but {1to }).C.Then 1 ∈ (), 1 ∈ and {1} ∈ . a)(a) False. Consider (1,not (b) 2 belongs to A and C but not to B. subset and |()| can only be a power of 2. b) True. Note that ⊂ means proper 3 belongs to A andexactly one of = B and c)(c) False. Consider = {} and {1}.C.Then () = {{}} and () = {{}, {1}}. 4 belongs to an number A, B and d)(d) True. There areeven only threeofsubsets of C. {1,2,3} with cardinality 2. ...
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