I'm having a couple confusions on this problem set. I got the hang of a lot of the notation and concepts, but I would really appreciate a look over to make sure I'm on the right track. My answers are in red, and problems that I'm a bit iffy on are in blue. Thanks in advance .
1. Given B = {m, n, p}, consider if the following statements are true or false.
a. m ∈ B
True
b. B ⊂ {m, n, p}
True
c. m ⊂ B
False
d. {m} ∈ B
False
e. {m} ⊂ B
True
d. ∅ ⊂ B
True
2. Given A = {2, {4, 5}, 4}, consider if the following statements are true or false.
a. {4, 5} ⊂ A
False
b. {{4, 5}} ⊂ A
True
c. {5} ⊂ A
False
d. {4, 5} ∈ A
True
e. {5} ∈ A
False
f. 5 ∈ A
False
3. Find all the subsets of the following sets
a. {1}
{1}, ∅
b. {a, b}
{a,b}, {a}, {b}, ∅
c. {{1, {3, 5}}}
{{1, {3, 5}}, {{1}}, {{3, 5}}, ∅
d. {1, 3, 5}
{1, 3, 5}, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, ∅
e. {{a}}
{{a}}, ∅
4. Find the power set for each of the following sets
a. {5}
P(a) = {{5}, ∅}
b. {0}
P(b) = {{0}, ∅}
c. {0, 1, 2}
P(c) = {{0}, {1}, {2}, {0,1}, {0, 2}, {1, 2}, {0, 1, 2}, ∅}
d. {{a, b}, c}
P(d) = {{{a, b}}, {c}, {{a, b}, c}, ∅}
e. ∅
P(e) = {∅}
f. P(∅)
P(f) = {∅}
5. Is {1, 3, 5, 7} a subset of {x: x is an odd positive number that is less than 10}, and if so, why?
Yes, because all elements in {1, 3, 5, 7} are members of the other set.
6. Consider if the following statements are true or false.
a. {4} ⊂ {{4}}
False
b. {4} ∈ {{4}}
True
c. ∅ ⊂ {{4}}
True
d. {0} = { }
False
e. ∅ ⊂ { }
True
f. ∅ = 0
False
g. 0 ∈ ∅
False
h. ∅ ∈ ∅
False
i. ∅ ∈ {∅}
True
J. ∅ ⊂ { }
True
7. Consider which of the following sets are subsets of set A
Let A = {x: x is a positive integer that is less than 50}
B = {x: x=2y, y is a positive integer that is less than 11}
C = {x: x=y2, y is an integer from 0 to 5}
D = {x: x is a positive odd number that is less than 50}
B ⊂ A, C ⊄ A, D ⊂ A
8. If A is a set with n numbers, find
a. The number of subsets of A having at least one member
2n-1
b. The number of subsets of A having n-1 members
n number of sets
9. Tell the number of proper subsets of a set with 5 members
2^5-1
31
1. Given B = {m, n, p}, consider if the following statements are true or false.
a. m ∈ B
True
b. B ⊂ {m, n, p}
True
c. m ⊂ B
False
d. {m} ∈ B
False
e. {m} ⊂ B
True
d. ∅ ⊂ B
True
2. Given A = {2, {4, 5}, 4}, consider if the following statements are true or false.
a. {4, 5} ⊂ A
False
b. {{4, 5}} ⊂ A
True
c. {5} ⊂ A
False
d. {4, 5} ∈ A
True
e. {5} ∈ A
False
f. 5 ∈ A
False
3. Find all the subsets of the following sets
a. {1}
{1}, ∅
b. {a, b}
{a,b}, {a}, {b}, ∅
c. {{1, {3, 5}}}
{{1, {3, 5}}, {{1}}, {{3, 5}}, ∅
d. {1, 3, 5}
{1, 3, 5}, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, ∅
e. {{a}}
{{a}}, ∅
4. Find the power set for each of the following sets
a. {5}
P(a) = {{5}, ∅}
b. {0}
P(b) = {{0}, ∅}
c. {0, 1, 2}
P(c) = {{0}, {1}, {2}, {0,1}, {0, 2}, {1, 2}, {0, 1, 2}, ∅}
d. {{a, b}, c}
P(d) = {{{a, b}}, {c}, {{a, b}, c}, ∅}
e. ∅
P(e) = {∅}
f. P(∅)
P(f) = {∅}
5. Is {1, 3, 5, 7} a subset of {x: x is an odd positive number that is less than 10}, and if so, why?
Yes, because all elements in {1, 3, 5, 7} are members of the other set.
6. Consider if the following statements are true or false.
a. {4} ⊂ {{4}}
False
b. {4} ∈ {{4}}
True
c. ∅ ⊂ {{4}}
True
d. {0} = { }
False
e. ∅ ⊂ { }
True
f. ∅ = 0
False
g. 0 ∈ ∅
False
h. ∅ ∈ ∅
False
i. ∅ ∈ {∅}
True
J. ∅ ⊂ { }
True
7. Consider which of the following sets are subsets of set A
Let A = {x: x is a positive integer that is less than 50}
B = {x: x=2y, y is a positive integer that is less than 11}
C = {x: x=y2, y is an integer from 0 to 5}
D = {x: x is a positive odd number that is less than 50}
B ⊂ A, C ⊄ A, D ⊂ A
8. If A is a set with n numbers, find
a. The number of subsets of A having at least one member
2n-1
b. The number of subsets of A having n-1 members
n number of sets
9. Tell the number of proper subsets of a set with 5 members
2^5-1
31
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