HW #7

Reminder: Each HW problem solution should consist of:

• a restatement of the entire problem (you may copy it word for word),
• followed by a complete solution with all intermediate steps shown.

All exercises are from §2.1 (sets).

1. Consider the following sets:
   1. {x ∈ R | x ∈ Z ∧ x < 1}
   2. {x ∈ R | (∃y∈Z)[x = y²]}
   3. {3, {3}}
   4. {{{3}}}
   1. (3 points each; total = 12 points)

      For each set S above, is 3 ∈ S?
      If the answer is "no", say why.

   2. (3 points each; total = 12 points)

      For each set S above, is {3} ∈ S?
      If the answer is "no", say why.

   3. (3 points each; total = 12 points)

      For each set S above, is {3} ⊆ S?
      If the answer is "no", say why.

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2. (3 points each; total = 24 points)

   Determine whether the following propositions are true or false.
   If a proposition is false, say why.

   1. 3 ∈ {3}
   2. 3 ⊆ {3}
   3. {3} ∈ {3}
   4. {3} ⊆ {3}
   5. 3 ∈ {3, {3}}
   6. 3 ⊆ {3, {3}}
   7. {3} ∈ {3, {3}}
   8. {3} ⊆ {3, {3}}

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3. (3 points)

   Find 2 sets S, T such that (S ∈ T) ^ (S ⊆ T)
   (note: more than one answer is possible).

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4. (3 points each; total = 12 points)

   For each set S below, what is |S|?

   1. S = {}
   2. S = {2}
   3. S = {{}, {3}}
   4. S = {4, {4}, {4, {4}}}

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5. (3 points each; total = 15 points)

   Let A = {a,b}; B = {1,2,3}; C = {i,j}. Compute the following Cartesian products:

   1. A X B
   2. B X A
   3. A X B X C
   4. A X A
   5. A X A X A

Total points = 90.

Tentative grading scheme:

A       86 - 90
A-      81 - 85
B+      76 - 80
B       71 - 75
B-      66 - 70
C+      61 - 65
C       51 - 60
C-      41 - 50
D+      31 - 40
D       16 - 30
F        0 - 15