**Unformatted text preview: **1.2. Let S = {−2, −1, 0, 1, 2, 3}. Describe each of the following sets as {x ∈ S : p(x)}, where p(x) is some
condition on x.
(a)
(b)
(c)
(d) A = {1, 2, 3}
B = {0, 1, 2, 3}
C = {−2, −1}
D = {−2, 2, 3} MATH 2345 section 54/55
Homework 4
Solutions 1.3. Determine the cardinality of each of the following sets: (a) A = {1, 2,
Q1. Let = #1, {13,},4,
({5}
1})*. Write × . How many elements does ( × ) have? (Don’t write
(b) B = {0, 2, 4, . . . , 20}
it down!)
(c) C = {25, 26, 27, . . . , 75}
(d) D = {{1, 2}, {1, 2, 3, 4}} ×=
(e) E = {∅}
((1,1), (1, {1}), /1, ({1})0, ({1} ,1), ({1}, {1}), /{1}, ({1})0, /({1}), 10, /({1}), {1}0, /({1}), ({1})0)
(f) F = {2, {2, 3, 4}}
()
= #∅, {1}, ({1}), #({1})* , (1, {1}), #1, ({1})* , #{1}, ({1})* , #1, {1}, ({1})* *
1.4. Write each of the following sets by listing its elements within braces.
5
|( (a)
× )|
A =={n2∈ Z : −4 < n ≤ 4} 5}
(b) B = {n ∈ Z : n2 <
< < < < 3
Q2. Let
= {n
#∈∈Nℝ|
≤ ≤ 7 * = >− 7 , 7 ? for all positive integers .
<
(c) C7 =
: n−
7 100} (d) D = {x ∈ R : x 2 − x = 0}
0}7
E =⋃
{x<B∈ R
x 2 +⋂1<B= a)(e)Find
7: and
7C< 7C< ⋃<B
7C< 7 = [−1,1] < < ⋂<B
7C< 7 = [− <B , <B] 1.5. Write each of the following sets in the form {x ∈ Z : p(x)}, where p(x) is a property concerning x.
G
G
b)(a)Find
7 and
A =⋃
{−1,
−3,⋂
. .7C<
.} 7
7C<−2, ⋃G
7C< 7 = [−1,1] ⋂G
7C< 7 = {0} (b) B = {−3, −2, . . . , 3} c)(c)Are
J , 1,
… 2}
mutually disjoint? No because ⋂G
< , I ,−1,
7C< 7 = {0}
C ={−2, 1.6. The set E = {2x : x ∈ Z} can be described by listing its elements, namely E = {. . . , −4, −2, 0, 2, 4, . . .}.
List the elements of the following sets in a similar manner. Q3. A group of college students were asked about their TV watching habits. Of those surveyed,
(a) A =watch
{2x + The
1 : xWalking
∈ Z}
28 students
Dead, 19watch The Blacklist, and 24 watch Game of Thrones.
n ∈ Z}The Walking Dead and The Blacklist, 14 watch The Walking Dead and
(b) B = {4n
Additionally,
16 :watch
= {3q + and
1 : q10
∈ Z}
Game(c)ofCThrones,
watch The Blacklist and Game of Thrones. There are 8 students who
watch
allset
three
How
surveyed
at leastbyone
of the
1.7. The
E =shows.
{. . . , −4,
−2,many
0, 2, 4,students
. . .} of even
integerswatched
can be described
means
of a shows?
defining condition
by E = {y = 2x : x ∈ Z} = {2x : x ∈ Z}. Describe the following sets in a similar manner. | ∪ ∪ |
(a) A = {. . . , −4, −1, 2, 5, 8, . . .}
= || + || + || − | ∩ | − | ∩ | − | ∩ | + | ∩ ∩ |
(b) B = {. . . , −10, −5, 0, 5, 10, . . .}
= 28 + 19 + 24 − 16 − 14 − 10 + 8 = 39
(c) C = {1, 8, 27, 64, 125, . . .} 1.8. Let A = {n ∈ Z : 2 ≤ |n|√< 4}, B =
Q4.
√ {x ∈ Q : 2 < x ≤ 4},
C = {x ∈ R : x 2 − (2 +
(a)
(b)
(c)
(d)
(e) 2)x + 2 2 = 0} and D = {x ∈ Q : x 2 − (2 + √
√
2)x + 2 2 = 0}. Describe the set A by listing its elements.
Give an example of three elements that belong to B but do not belong to A.
Describe the set C by listing its elements.
Describe the set D in another manner.
Determine the cardinality of each of the sets A, C and D. (c) A ∈ B and A ⊂ C.
(c) Describe the set C by listing its elements.
1.11. Let (a, b) be an open interval of real numbers and let c ∈ (a, b). Describe an open interval I centered at c
such
I ⊆ (a,
(d) that
Describe
theb).
set D in another manner. 1.12. Which
of the following
sets are equal?
(e) Determine
the cardinality
of each of the sets A, C and D.
A = {n ∈ Z : |n | < 2}
D = {n ∈ Z : n 2 ≤ 1}
B = {n ∈ Z : n 3 = n }
E = {−1, 0, 1}.
Solution.
C = {n ∈ Z : n 2 ≤ n }
(a) The set A listed explicitly is A = {−3, −2, 2, 3}.
1.13. For a universal set U = {1, 2, . . . , 8} and two sets A = {1, 3, 4, 7} and B = {4, 5, 8}, draw a Venn diagram
(b) represents
Since the these
elements
that
sets.in B are rational numbers, any three rational numbers
(that are not simultaneously integers) in B will not be in A. Three such
1.14. Find P(A) and |P(A)|
for
examples are: 52 , 27 and 13
4 .
(a) A = {1, 2}.
√
√
√
(c) ANote
that
x2 − (2 + 2)x +√
2 2 = 0 can be written as (x − 2)(x − 2) = 0.
(b)
= {∅,
1, {a}}.
This implies that C = {2, 2}.
1.15. Find P(A) for A = {0, {0}}.
(d) Since D is restricted to Q, we have D = {2}.
1.16. Find P(P({1})) and its cardinality.
(e) P(A)
The cardinalities
are:A |A|
= 4,
∅, |C|
{∅}}.= 2, and |D| = 1.
1.17. Find
and |P(A)| for
= {0,
1.18. For A = {x : x = 0 or x ∈ P({0})}, determine P(A). 1.19. Remark.
Give an example
set S such
that
There of
area infinite
possibilities
for part (b). For part (c), we could
have
optionally
used
the
quadratic
formula
if the factorization is not readily
(a) S ⊆ P(N)
recognized. As for set D, note that Z ⊂ Q, so it is not necessary to write 2 as
(b) S ∈ P(N)
a fraction.
(c) S ⊆ P(N) and |S | = 5
(d) S ∈ P(N) and |S | = 5 1.20.
Q5. Determine whether the following statements are true or false.
(a)
(b)
(c)
(d) If {1} ∈ P(A), then 1 ∈ A but {1} ∈
/ A.
If A, B and C are sets such that A ⊂ P(B) ⊂ C and |A| = 2, then |C| can be 5 but |C| cannot be 4.
If a set B has one more element than a set A, then P(B) has at least two more elements than P(A).
If four sets A, B, C and D are subsets of {1, 2, 3} such that |A| = |B| = |C| = |D| = 2, then at least
two of these sets are equal. 1.21. Three subsets A, B and C of {1, 2, 3, 4, 5} have the same cardinality. Furthermore, 1 belongs
to A andB=but
{1to
}).C.Then 1 ∈ (), 1 ∈ and {1} ∈ .
a)(a) False.
Consider
(1,not
(b)
2
belongs
to
A
and
C
but
not
to
B. subset and |()| can only be a power of 2.
b) True. Note that ⊂ means proper
3 belongs
to A andexactly
one of =
B and
c)(c) False.
Consider
= {} and
{1}.C.Then () = {{}} and () = {{}, {1}}.
4 belongs
to an
number
A, B and
d)(d) True.
There
areeven
only
threeofsubsets
of C.
{1,2,3} with cardinality 2. ...

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