HW #8

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 §2.2 (set operations).

1. (3 points each; total = 12 points)

   Let A = {a, b, c, d, e} and B = {a, e, i, o, u}. Find

   1. A ∪ B
   2. A ∩ B
   3. A – B
   4. B – A

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2. (6 points each; total = 12 points)

   (This is problem 16d, e on p. 131.)
   Let A, B be sets. Prove each of the following, justifying each step.

   1. A ∩ (B – A) = ∅
   2. A ∪ (B – A) = A ∪ B

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3. (3 points each; total = 6 points)

   Let A, B be sets.
   Then the symmetric difference of A and B (denoted A ⊕ B)  =_def the set containing those elements in either A or B, but not in both A and B.

   1. Express this definition in the language of first-order predicate logic plus set theory;
      i.e., find a predicate P such that A ⊕ B =_def {x | P(x)}.

   2. Find {a, c, e} ⊕ {a, b, c}.

   ---

4. (3 points each, total = 12 points; for full credit, you must show your work, not just your answer.)

   Hint: Compute A₀, A₁, A₂, A₃, …, to see what patterns you can find that might help you compute the answers.

   Find ∪_i ∈ N A_i and ∩_i ∈ N A_i, if, for every i ∈ N,

   1. A_i = {i, 2i, 3i, …}.
   2. A_i = {i, i², i³, …}

Total points = 42.

Tentative grading scheme:

A       41 - 42
A-      38 - 40
B+      36 - 37
B       34 - 35
B-      31 - 33
C+      29 - 30
C       24 - 28
C-      20 - 23
D+      15 - 19
D        8 - 14
F        0 -  7