Feb:13(Fri)

MTH 225 Exercise-Set #5

Due February 20 (Friday)


    Acknowledgement: Some of these exercises are derived from Richard Johnsonbaugh.  (Some of these exercises are derived from Kenneth Rosen.)

  1. [8 points]   For each of the following sets which are specified in this exercise using "set builder" notation, do the following: Here are the specifications of sets for this exercise:
    1.   { x : x is a binary digit }
    2.   { x : x is a letter in your last name }
      (For a team of two students doing this work, it'll suffice to handle the set for only one of your last names.)  Use lowercase for every letter.
    3.   { x : x ∈ N and x² < 20 }
    4.   { x : x = 2 or x = 7 }
    5.   { x : x = 2 and x = 7 }
    6.   { x + y :  x = 2 or x = 7  and  y = 2 or y = 7  }
    7.   { S : S is the set of instructors of MTH 225 this semester
                or S is the set of instructors of CS 163 this semester }
                or S is the set of instructors of CS 361 this semester }
      (Rem. you can use GVSU's online registration system Banner to determine lists of instructors teaching various courses.)
    8.   { S : S is a set containing two letters of "helo" }
    OK, there isn't much intermediate work in this exercise; so you don't need to worry about showing such here.

  3. [3 points]   Tell whether each of the following relationships is true or false:
    1.   {x} ∈ {x}
    2.   {x} ∈ {x,{x}}
    3.   {x} ⊂ {x,{x}}
    4.   {x} ⊆ {x,{x}}
    Include brief explanations of your answers.

  4. [2 points]   (2.1.18)   Give the values of the following expressions:
    1.   |∅|
    2.   | { ∅ } |
    3.   | { ∅, {∅} } |
    4.   | { ∅,  {∅},  {∅, {∅}} } |
    (Consider using ProofBuilder for the initial items here.  Except if the version of ProofBuilder that you have returns 0 for item (ii) here, then try downloading a more current version.)

  5. [2 points]  
    1. If set X has 10 members, then how many members does ℘(X) have?
    2. How many proper subsets does X have?
    Show intermediate work/explanations for obtaining answers.

  6. [8 points]   Let A := {1,2}, B := {a,b,c}, C := {α,β}, and D := {$}.  Give list representations of each of the following sets.  Be careful to use correct notation (appropriate parentheses, braces).
    1.   A × B
    2.   A × A
    3.   A × C × D
    4.   A × D × D
    5.   A × A × A

  7. [10 points]   Let the universe be the set U := {1,2,3,4,5,6,7,8,9,10}.  Let A := {1,4,7,10}, B := {1,2,3,4,5}, and let C := {2,4,6,8}. 

    1. [3 points] Draw a Venn diagram for this universe U and these sets A,B,C with the elements 1,2,3,4,5,6,7,8,9,10 placed appropriately in the diagram.

    2. Concisely list the elements of the sets produced by the following operations on the sets A,B,C — except write "∅" instead of "{ }" for any set that is empty.
      1.   B ∩ C
      2.   A – B
      3.   A   a.k.a.   ~A   a.k.a.   Ac   a.k.a.   A′   a.k.a.   A'
      4.   U – C
      5.   A ∪ ∅
      6.   B ∩ ∅
      7.   B ∩ U
      8.   A ∩ (B ∪ C)
      9.   (A ∩ B) – C
      10.   (A ∩ B) ∪ C     a.k.a.   [~(A ∩ B)] ∪ C   a.k.a.   (A ∩ B)c ∪ C   a.k.a.   (A ∩ B)′ ∪ C   a.k.a.   (A ∩ B)' ∪ C
      Show intermediate work as appropriate, notably for [viii], [ix], and [x].

    1. [2 points]   'Flesh out' the back-of-the-book solutions of our textbook's Exercise 2.1.33c,d.
      Use ProofBuilder.  Note that by contrast with our textbook, ProofBuilder requires typing parentheses around 'complicated' quantifiers such as the ones here. 

    2. [2 points]   Do our textbook's Exercise 2.1.34a,b.
      Use ProofBuilder.  Note that by contrast with our textbook, ProofBuilder requires typing parentheses around 'complicated' quantifiers such as the ones here. 

  9. [3 points]   Confirm each distributivity for our textbook's Exercise 2.2.25c,d like in lecture, showing intermediate sets....

  10. [6 points]   For each of the following cases of different collections of sets Si, give concise(!) list representations of  ∪Si  and  ∩Si  — except write "∅" instead of "{ }" for any set that is empty.
    1.   S1 := {1,3,9}, S2 := {1,3,5}, and S3 := {1,2,3,6}
    2.   S1 := {w,h,i,s,p,e,r}, S2 := {t,h,i,s}, and S3 := {q,u,i,c,k,,l,y}
    3.   S1 := the set of letters of your full first name, S2 := the set of letters of your full middle name, and S3 := the set of letters of your full last name.  Use lowercase for every letter.
      For a team of two students doing this work, it'll suffice to do this work for only one of you.  Show S1, S2, and S3 here.

  11. Suppose there's a group of 191 students, of which 10 are taking French and business and music, 36 are taking French and business, 20 are taking French and music, 18 are taking business and music, 65 are taking French, 76 are taking business, and 63 are taking music.
    Let the universe U be the set of all these students, let B be the set of students taking business, let F be the set of students taking French, and let M be the set of students taking music.
    1. [9 points]   Produce eight separate Venn diagrams using shading to depict each of the eight (sub)groups specified above, with labels which are expressions using the symbols F, B, M and set operators and cardinalities and the sizes.  (In case it's not clear, note that you need to include the overall group (specified first).)  For example, for the subgroup taking French and business, a Venn diagram with the relevant label is as follows:
                |F ∩ B| = 36
      (so you need to include a labeled diagram like this among your answers ;-) .
    2. [8 points]   Give five expressions and draw five separate Venn diagrams using shading for the following sets (using the symbols F, B, M and set operators):
      1. the set of students taking French or business (or both)
      2. the set of students taking French and music but not business
      3. the set of students taking music or French (or both) but not business
      4. the set of students taking business and neither French nor music
      5. the set of students taking none of the three subjects
      (Don't worry about the cardinalities of these sets — for now. ;-)

    1. [2 points]   Which of these collections of sets are partitions of {1,2,3,4,5,6}?
      1.   { {1,2}, {2,3,4}, {4,5,6} }
      2.   { {1}, {2,3,6}, {4}, {5} }
      3.   { {2,4,6}, {1,3,5} }
      4.   { {1,4,5}, {2,6} }
      Explain when a collection is not a partition.

    2. [2 points]   Which of these collections of sets are partitions of {-3,-2,-1,0,1,2,3}?
      1.   { {-3,-1,1,3}, {-2,0,2} }
      2.   { {-3,-2,-1,0}, {0,1,2,3} }
      3.   { {-3,3}, {-2,2}, {-1,1}, {0} }
      4.   { {-3,-2,2,3}, {-1,1} }
      Explain when a collection is not a partition.

  13. Here are some axioms and identities which you can use: 
    ∀S1∀S2[S1 ⊆ S2  ↔  ∀x(x ∈ S1  →  x ∈ S2)]
∀S1∀S2[S1 = S2  ↔  ∀x(x ∈ S1  ↔  x ∈ S2)]
∀S1∀S2[S1 = S2  ↔  (S1 ⊆ S2 ∧ S2 ⊆ S1)]
∀S1∀S2∀x[x ∈ S1 ∪ S2  ↔  (x ∈ S1  ∨  x ∈ S2)]
∀S1∀S2∀x[x ∈ S1 ∩ S2  ↔  (x ∈ S1  ∧  x ∈ S2)]
∀S1∀S2∀x[x ∈ S1 - S2  ↔  (x ∈ S1  ∧  x ∉ S2)]
∀S∀x[x ∈ ~S  ↔  x ∉ S]

∀S1∀S2[~(S1 ∩ S2) = ~S1 ∪ ~S2]
∀S1∀S2[S1 - S2 = S1 ∩ ~S2]
∀S1∀S2∀S3[S1 ∩ (S2 ∪ S3) = (S1 ∩ S2) ∪ (S1 ∩ S3)]
    Using ProofBuilder, produce proofs of the formulas below.  Use the axioms and identities above — and further, in your proofs of the numbered formulas below, you are allowed to use lower-numbered formulas.  Thus, the proof of Example 14 of page 126 used 'earlier' formulas, not just axioms.

    Rem. you can copy material from here and lecture notes and paste such into ProofBuilder.

    1. ∀S[∅ ⊆ S]
      See the manual for ProofBuilder — not to mention our textbook's Theorem 1 on page 115.

    2. ∀S[S ⊆ S]

    3. A ⊆ B ∧ B ⊆ C  →  A ⊆ C     (2.1.15)

    4. ∀S[S ∪ S = S]

    5. ∀S[∅ ∪ S = S]

    6. ∀S[S ∩ ∅ = ∅]

    7. ∀S1∀S2[S1 ∪ (S1 ∩ S2) = S1]

    8. ∀S[~~S = S]

    9. ∀S[S ∩ ~S = ∅]

    10. ∀S1∀S2[S1 ∩ (S2 - S1) = ∅]

    11. A - (A - B) = A ∩ B
      This is #5 among the supplementary exercises at the end of Chapter 2 of our textbook.  Note the solution in the back.