MAT 299 Module 3 HW.docx
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Course
299
Subject
Mathematics
Date
Apr 3, 2024
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MAT 299 Module Three Homework
General:
●
Before beginning this homework, be sure to read the textbook sections and the material
in Module Three.
●
Type your solutions into this document and be sure to show all steps for arriving at your
solution. Just giving a final number may not receive full credit.
●
You may copy and paste mathematical symbols from the statements of the questions
into your solution. This document was created using the Arial Unicode font.
●
These problems are proprietary to SNHU COCE, and they may not be posted on any
non-SNHU website.
●
The Institutional Release Statement in the course shell gives details about SNHU’s use
of systems that compare student submissions to a database of online, SNHU, and other
universities’ documents.
Duvall Roberts
SNHU MAT299
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1
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4
Module Three Homework
1.
Are these statements true or false? The universe of discourse is the set of all people, and
T(x, y) means “x and y are twins.” Explain how you determined your solution.
a.
∀
x ¬T(x, x)
No one is their own twin. True. By the definition of twins, an individual cannot be
their own twin since twins involve two distinct individuals.
b.
∃
y T(y, y)
There exists someone who is their own twin. False. As stated above, an individual
cannot be their own twin.
c.
∃
x
∀
y T(x, y)
There exists someone who is twins with everyone. False. It is not logically possible
for one person to be twins with every other person.
d.
∀
x ¬
∃
y T(x, y)
No one has a twin. False. This statement implies that twins do not exist, which is not
true.
e.
∃
x ¬
∃
y T(x, y)
There exists someone who does not have a twin. True. There certainly are
individuals who do not have a twin.
f.
∀
x ¬
∀
y T(x, y)
Not everyone is twins with everyone else. True. This statement is logically true since
not everyone can be twins with each other.
This is similar to
examples and exercises
in Section 2.1 of your SNHU MAT299 textbook.
2.
Are these statements true or false? The universe of discourse is ℕ. Explain how you
determined your solution.
a.
∀
x(5 < x < 10 →
∃
a
∃
b
∃
c(a
2
+ b
2
+ c
2
= x)).
True. For integers 6, 7, 8, and 9, you can find integers a,b,c such that their squares
sum up to x.
b.
∃
!x((x – 4)
2
= 36).
False. The equation (x – 4)
2
= 36 has two solutions: x = 10 and x = -2. However,
since the universe is ℕ x = -2 is not valid, leaving x = 10 as the only solution in ℕ,
making the statement true in the context of natural numbers.
c.
∃
!x((x – 11)
2
= 49).
False. The equation (x – 11)
2
= 49 has two solutions: x = 18 and x = 4, both of which
are in ℕ.
d.
∃
x
∃
y ( (x ≠ y)
∧
((x – 4)
2
= 36)
∧
((y – 4)
2
= 36) ).
False. The equation (x – 4)
2
= 36 can only yield two solutions, and in ℕ, only, x = 10
is valid. Therefore, there cannot be a distinct y in ℕ that satisfies the same condition
without being equal to x.
This is similar to
examples and exercises
in Section 2.2 of your SNHU MAT299 textbook.
SNHU MAT299
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Module Three Homework
3.
Show that ¬
∀
x
∈
A ¬P(x) is equivalent to
∃
x
∈
A P(x).
¬
∀
x
∈
A ¬P(x) means "it is not true that for all x in A, P(x) is false". This logically implies
that there must be at least one x in A for which P(x) is true, which is exactly what
∃
x
∈
A
P(x) states.
This is similar to the examples
and exercises
in Section 2.2 of your SNHU MAT299
textbook.
4.
Analyze the logical forms of the following statements. The universe of discourse is ℝ.
a.
The identity element for multiplication is 1.
∀
x
∈
ℝ, x
⋅
1=1 The statement asserts that multiplying any real number by 1 leaves it
unchanged, which defines 1 as the identity element for multiplication in the real
numbers.
b.
Every positive real number has a positive multiplicative inverse.
∀
x > 0,
ヨ
y > 0(x
⋅
y=1) This statement means for every positive real number, there
exists another positive real number such that their product is 1, defining the concept
of a multiplicative inverse.
c.
No positive real number has a negative multiplicative inverse.
∀
x > 0, ¬
ヨ
y < 0(x
⋅
y=1) This shows that for any positive real number, there does not
exist a negative real number that when multiplied by the positive number gives 1,
reinforcing the definition of a multiplicative inverse within the positives.
This is similar to
examples and exercises
in Section 2.2 of your SNHU MAT299 textbook.
5.
Let J = {2, 3, 4}, and for each j
∈
J, let A
j
= {j, j + 1, j + 2, 2j, 3j}.
a.
List the elements of all the sets A
j
, for j
∈
J. Explain how you determined these sets.
= {2,3,4,6}
𝐴
2
= {3,4,5,6,9}
𝐴
3
= {4,5,6,8,12}
𝐴
4
This method systematically expands each set based on its defined rule, ensuring a
structured approach to determining the elements.
b.
Find ∩
j
∈
J
A
j
and
∪
j
∈
J
A
j
. Explain how you determined these sets.
Intersection: ∩
j
∈
J
A
j
=
= {4,6}
𝐴
2
∩𝐴
3
∩𝐴
4
SNHU MAT299
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Module Three Homework
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Union:
∪
j
∈
J
A =
= {2,3,4,5,6,8,9,12}
𝐴
2
∪𝐴
3
∪𝐴
4
The intersection ∩j
∈
J Aj finds elements common to all
sets, which requires
𝐴
𝑗
identifying numbers that adhere to the rules of all three sets. The union
∪
j
∈
J A
combines all unique elements from each
creating a set that encapsulates the
𝐴
𝑗
entire range of generated numbers. The process involves systematic comparison
and combination of elements from each
to identify commonalities and aggregate
𝐴
𝑗
all unique values.
This is similar to
examples and exercises
in Section 2.3 of your SNHU MAT299 textbook.
6.
Show the following for any two sets A and B.
a.
℘(A)
∪
℘(B)
⊆
℘(A
∪
B).
True. Any subset of A or B is also a subset of A
∪
B, hence their power sets' union
is a subset of the power set of A
∪
B.
b.
Is ℘(A)
∪
℘(B) = ℘(A
∪
B)? Either provide a proof to show that this is true or
provide a counterexample to show that this is false.
No. Consider A = {1} and B = {2}. ℘(A)
∪
℘(B) = {{},{1},{2}}, but ℘(A
∪
B) =
{{},{1},{2},{1,2}}. The set {1,2} makes them different.
This is similar to
examples and exercises
in Section 2.3 of your SNHU MAT299 textbook.
SNHU MAT299
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Module Three Homework