Difusion Conjunto

CSE260


Solutions to Homework Set #5

1. List the members of these sets.
a) {x | x is a real number such that x² = 1}
{-1, 1}
b) {x | x is a positive integer less than 12}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
c) {x | x is the square of an integer and x < 100}
{0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
d) {x | x is an integer such that x² = 2}
∅

3. Determine whether each of thesepairs of sets are equal.
a) {1, 3, 3, 3, 5, 5, 5, 5, 5}, {5, 3, 1}
Yes
b) {{1}}, {1, {2}}
No
c) ∅, {∅}
No

7. Determine whether these statements are true or false.
a) 0 ∈ ∅
False
b) ∅ ∈ {0}
False
c) {0} ⊂ ∅
False
d) ∅ ⊂ {0}
True
e) {0} ∈ {0}
False
f) {0} ⊂ {0}
False
g) {∅} ⊆ {∅}
True
11. Use a Venn diagram to illustrate the set of all months of the year whosenames do not contain the letter R in the set of all months of the year.

















13. Use a Venn diagram to illustrate the relationships A ⊂ B and B ⊂ C.






















17. What is the cardinality of each of these sets?
a) {a}
1
b) {{a}}
1
c) {∅, {∅}}
2
d) {a, {a}, {a, {a}}}
3

25. What is the Cartesian product A x B x C, where Ais the set of all airlines and B and C are both the set of all cities in the United States?
The set of triples (a, b, c), where a is an airline and b and c are cities.

27. Let A be a set. Show that ∅ x A = A x ∅ = ∅.
∅ x A = {(x, y) | x ∈ ∅ and y ∈ A} = ∅ = {(x, y) | x ∈ A and y ∈ ∅} = A x ∅

38. This exercise presents Russell's paradox. Let S be the set that contains a set x if theset x does not belong to itself, so that S = {x | x ∈ x}.
a) Show the assumption that S is a member of S leads to a contradiction.
If S ∈ S, then by the defining condition for S we conclude that S ∉ S, a contradiction.
b) Show the assumption that S is not a member of S leads to a contradiction.
If S ∉ S, then by the defining condition for S we conclude that it is not the case that S ∉ S(otherwise S would be an element of S), again a contradiction.

39. Describe a procedure for listing all the subsets of a finite set.
Let S = {a1, a2, ..., an}. Represent each subset of S with a bit string of length n, where the ith bit is 1 if and only if ai ∈ S. To generate all subsets of S, list all 2^n bit strings of length n (for instance, in increasing order), and write down thecorresponding subsets.

1. Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets.
a) A ∩ B
The set of students who live within one mile of school and who walk to classes.
b) A ∪ B
The set of students who live within one mile of school or who walk to classes.
c) A – B
The set of studentswho live within one mile of school but do not walk to classes.
d) B – A
The set of students who walk to classes but live more than one mile away from school.
3. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find
a) A ∪ B
{0, 1, 2, 3, 4, 5, 6}
b) A ∩ B
{3}
c) A – B
{1, 2, 4, 5}
d) B – A
{0, 6}

16. Let A and B be sets. Show that
a) (A ∩ B) ⊆ A.
If x is in A ∩ B, then perforce itis in A (by definition of intersection).
d) A ∩ (B – A) = ∅.
If x ∈ A then x ∉ B – A. Therefore there can be no elements in A ∩ (B – A), so A ∩ (B – A) = ∅.

29. What can you say about the sets A and B if we know that
a) A ∪ B = A?
B ⊆ A
b) A ∩ B = A?
A ⊆ B
e) A – B = B – A?
A = B

33. Find the symmetric difference of the set of computer science majors at a school and the setof mathematics majors at this school.
The set of students who are computer science majors but not mathematics majors or who are mathematics majors but not computer science majors.

49. Find ∪i=1,∞ Ai and ∩i=1,∞ Ai if for every positive integer i,
a) Ai = { –i, –i + 1, ..., -1, 0, 1, ..., i – 1, i}.
Z, {-1, 0, 1}
d) Ai = [i, ∞], that is, the set of real numbers x with x ≥ i.
[1, ∞), ∅...