Discrete Structures

HW #7

 Last Update: 16 March 2009 Note: or material is highlighted

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. {xR | xZx < 1}
2. {xR | (∃yZ)[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.

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}}

3. (3 points)

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

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}}}

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.

```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
```

 DUE AT BEGINNING OF LECTURE, THIS FRIDAY, MARCH 20!!