I have a worksheet on Induction and Relations and functions that I could use some help with. I need to have explanations for the answers/ work shown, please. I want to compare to what I think is correct. For question #16 & #17 I have included the pages it refers to.ThanksPatrickmath301week2homework.docxmath301week2homework.pdfweek_2_ordered_fields.pdfMATH 301 Week 2 Homework, Spring, 2016. Due Sunday, February 14.
NAME: _____________________________________
Instructions: You are encouraged to discuss homework, use online resources, and to seek help from the instructor when you
need it, but your submitted write-up of your work must be your own, in your own words.
Induction
#1. Prove by induction that 9 + 12 + 15 + … + 3(n + 2) = 3n(n + 5)/2 for all n N.
Page 1 of 8
#2. Prove the summation formula for the following particular finite geometric series. (Do not prove the
general formula for a geometric progression and then substitute the value for the ratio. Instead, prove this
particular formula shown below.)
Prove by induction that
4
1 2
1
1
+ 4( ) + ⋯+ 4( ) = 1 −
5
5
5
5
for every n N.
#3. Prove by induction that (1.01)n 1 + (0.01)n for every n N .
Page 2 of 8
Relations and Functions
#4. Find an example of a relation that is reflexive and symmetric, but not transitive. (Note that you can
just list an appropriate set of ordered pairs.)
#5. Let S be a set. Let A and B be subsets of S. Let R be the relation given by A R B iff A B.
That is, A is related to B iff A is a subset of B.
State which of the three properties (reflexive, symmetric, and transitive) apply. If a property does not
hold, provide a counterexample.
#6. State the range of each function f: R → R. (no work required to be shown)
#6(a) f(x) = 3 sin(4x)
#6(b) f(x) = 2(x + 1)2 + 5
#7. Let X = {1, 2, 3, 4, 5} and Y = {4, 5 ,6, 7}. Define a function f: X → Y as follows:
f (1) = 7, f (2) = 6, f (3) = 4, f (4) = 5, f (5) = 6.
For each statement, is it True or False? (no explanation required)
________ (a) ∃ x ∈ X such that f (x) = 3x.
________ (b) ∃ y ∈ Y such that ∀ x ∈ X, f (x) = y.
________ (c) ∀x ∈ X, ∃ y ∈ Y such that f (x) = y.
________ (d) ∀y ∈ Y, ∃ x ∈ X such that f (x) = y.
________ (e) f is injective.
________ (f) f is surjective.
#8. Let f: A → B. Recall that f is injective iff a, a’ A, if f(a) = f(a’), then a = a’.
Write the contrapositive of the quantified if-then statement (the statement in bold).
Page 3 of 8
#9. Claim: The function f: R → R defined by f(x) = 7 − 4x is injective. Consider the following “proofs” of
the claim. INSTRUCTIONS: Critique each proof (A, B, C, D, E). For each proof, is it a valid argument
establishing the claim or not? What are the flaws, if any?
Proof A:
Let a = 0 and a’ = 1.
f(a) = 7 and f(a’) = 3.
Since f(a) f(a’ ) and a a’ , f is injective.
Proof B:
Let a, a’ R and suppose a = a’.
Multiply both sides by −4, so −4a = −4a’
Add 7 to both sides, so
7 − 4a = 7 − 4a’
Thus, f(a) = f(a’).
Therefore f is injective.
Proof C:
Let a, a’ R and suppose a a’.
Then
−4a −4a’
and also 7 − 4a 7 − 4a’
So, f(a) f(a’).
Therefore f is injective.
Proof D:
Let a, a’ R and suppose f(a) = f(a’).
Then 7 − 4a = 7 − 4a’.
Subtract 7 from both sides, so −4a = −4a’ .
Divide both sides by −4, so a = a’.
Thus f is injective.
Proof E:
Let a, a’ R and suppose f(a) f(a’).
Then 7 − 4a 7 − 4a’
and
−4a −4a’
and so
a a’
Thus f is injective.
Page 4 of 8
#10. Define f: R → R defined by f(x) = x2. Let S = [0, 9] and T = [− 9, 0].
#10(a) Find f(S), f(T), and f(S T). Is f(S T) = f(S) f(T)?
#10(b) Find f −1(S), f −1 (T), and f −1 (S T). Is f −1 (S T) = f −1 (S) f −1 (T)?
#11. Classify each function as injective, surjective, bijective, or none of these, as appropriate. If not
injective, briefly explain. If not surjective, briefly explain. (Otherwise, no explanation required.)
#11 (a) f: Z → Z defined by f(n) = 5 − n
#11 (b) f: N → Z defined by f(n) = n2 + 2n
#11 (c) f: Z → Z defined by f(n) = n2 + 2n
Page 5 of 8
Cardinality
#12. For each statement, is it True or False? (no explanation required)
1
1
1
1
________ (a) ⋂∞
=1 (6 − , 6 + ) is a countable set.
________ (b) ⋃∞
=1 (6 − , 6 + ) is a countable set.
________ (c) No subset of the irrational numbers is countable.
________ (d) Every subset of the rational numbers is countable.
#13. State a specific function f that is a bijection f: (0, 1) → (1, ). (Note that you can then conclude that
intervals (0, 1) and (1,) have the same cardinality. You are not asked to carry out a formal proof that
your f is bijective.)
#14. HINT: Look at page 2 of my posted notes on Cardinality.
#14(a) Show that the intervals (0, 1) and (3, 8) have the same cardinality by finding a
bijection f:(0, 1) → (3, 8).
#14(b) Suppose r < s.
Prove that the intervals (0, 1) and (r, s) have the same cardinality by finding a bijection f:(0, 1) → (r, s).
#14(c) Suppose a < b and c < d. Our goal is to show that open intervals (a, b) and (c, d) have the same
cardinality. By part (b), there exist bijections g:(0, 1) → (a, b) and h:(0, 1) → (c, d).
State a specific function f that is a bijection f: (a, b) → (c, d), where f is an appropriate composition of
functions involving functions g, h, and/or their inverses. [Recall composition of functions ---see Relations and
Functions notes, page 9]. You do not need a complicated formula. Just make use some of the functions g,
h, g−1, h−1, with an appropriate composition.
Page 6 of 8
ORDERED FIELDS
#15. Consult the list of properties A1 - A5, M1 - M5, DL, O1 - O4 from my Ordered Field notes.
Rather than considering those properties applied on the set R of real numbers, restrict the set as
indicated below. In other words, check which properties still apply, when R is replaced by the set
specified.
#15 (a) Which properties are not satisfied on the set N? (Just list the identifying labels.)
#15(b) Which properties are not satisfied on the set Z? (Just list the identifying labels.)
#15(c) Let S = {x in R such that x 0}. Which properties are not satisfied on the set S? (Just list the
identifying labels.)
#16. Prove that 0 < 1/2 < 1. Fill in the blanks of the proof. Refer to the field axioms and order axioms
and the Theorem in my Ordered Field notes, pages 1-2.
Proof:
Note that 0 < 1 (by Theorem part ___)
Adding 1 to both sides, 0 + 1 < 1 + 1 by order axiom ___.
0 + 1 = 1 by field axiom _____, and 1 + 1 = successor of 1, which is designated by 2 (Peano axiom in
Induction notes).
So, we have 1 < 2.
Since 0 < 1 and 1 < 2, we have 0 < 1 < 2.
Then 0 < 1/2 < 1/1 by Theorem, part ___
Note that 1/1 = 1 (because 1 1 = 1).
Thus, we have 0 < 1/2 < 1, as desired.
Page 7 of 8
#17. Let x be a real number.
Claim: If 0 x < for all real numbers > 0, then x = 0.
Fill in the blanks of the proof of the claim. Refer to the field axioms and order axioms and the Theorem
in my Ordered Field notes, pages 1-2.
Proof of Claim, by contradiction:
Suppose not.
Suppose for all real numbers > 0, we have 0 x < , but x 0.
Since 0 x and x 0, we must have x __ 0. (fill in blank with , whichever is appropriate)
Set = (1/2) x.
Since 1/2 > 0 (by problem #16) ,
we have = (1/2) x > (1/2) 0 by order axiom ____.
=0
since (1/2) 0 = 0 by Theorem, part ___.
Since 1/2 < 1 (by problem #16) and x __ 0, we have = (1/2) x < 1 x by order axiom ____,
and so = (1/2) x < x
since 1 x = x by field axiom ____.
In summary, we know x __ 0 and we have found a particular > 0 for which x __ . (fill in the blanks to
show the correct order relationships between the numbers)
We have reached a contradiction of our hypothesis that x < , and so we conclude that x must be equal
to 0.
Page 8 of 8
MATH 301 Week 2 Homework, Spring, 2016. Due Sunday, February 14.
NAME: _____________________________________
Instructions: You are encouraged to discuss homework, use online resources, and to seek help from the instructor when you
need it, but your submitted write-up of your work must be your own, in your own words.
Induction
#1. Prove by induction that 9 + 12 + 15 + … + 3(n + 2) = 3n(n + 5)/2 for all n ∈ N.
Page 1 of 8
#2. Prove the summation formula for the following particular finite geometric series. (Do not prove the
general formula for a geometric progression and then substitute the value for the ratio. Instead, prove this
particular formula shown below.)
Prove by induction that
+ 4 + ⋯+ 4 = 1 −
for every n ∈ N.
#3. Prove by induction that (1.01)n ≥ 1 + (0.01)n for every n ∈ N .
Page 2 of 8
Relations and Functions
#4. Find an example of a relation that is reflexive and symmetric, but not transitive. (Note that you can
just list an appropriate set of ordered pairs.)
#5. Let S be a set. Let A and B be subsets of S. Let R be the relation given by A R B iff A ⊆ B.
That is, A is related to B iff A is a subset of B.
State which of the three properties (reflexive, symmetric, and transitive) apply. If a property does not
hold, provide a counterexample.
#6. State the range of each function f: R → R. (no work required to be shown)
#6(a) f(x) = 3 sin(4x)
#6(b) f(x) = 2(x + 1)2 + 5
#7. Let X = {1, 2, 3, 4, 5} and Y = {4, 5 ,6, 7}. Define a function f: X → Y as follows:
f (1) = 7, f (2) = 6, f (3) = 4, f (4) = 5, f (5) = 6.
For each statement, is it True or False? (no explanation required)
________ (a) ∃ x ∈ X such that f (x) = 3x.
________ (b) ∃ y ∈ Y such that ∀ x ∈ X, f (x) = y.
________ (c) ∀x ∈ X, ∃ y ∈ Y such that f (x) = y.
________ (d) ∀y ∈ Y, ∃ x ∈ X such that f (x) = y.
________ (e) f is injective.
________ (f) f is surjective.
#8. Let f: A → B. Recall that f is injective iff ∀ a, a' ∈ A, if f(a) = f(a'), then a = a'.
Write the contrapositive of the quantified if-then statement (the statement in bold).
Page 3 of 8
#9. Claim: The function f: R → R defined by f(x) = 7 − 4x is injective. Consider the following "proofs" of
the claim. INSTRUCTIONS: Critique each proof (A, B, C, D, E). For each proof, is it a valid argument
establishing the claim or not? What are the flaws, if any?
Proof A:
Let a = 0 and a' = 1.
f(a) = 7 and f(a') = 3.
Since f(a) ≠ f(a' ) and a ≠ a' , f is injective.
Proof B:
Let a, a' ∈ R and suppose a = a'.
Multiply both sides by −4, so −4a = −4a'
Add 7 to both sides, so
7 − 4a = 7 − 4a'
Thus, f(a) = f(a').
Therefore f is injective.
Proof C:
Let a, a' ∈ R and suppose a ≠ a'.
Then
−4a ≠ −4a'
and also 7 − 4a ≠ 7 − 4a'
So, f(a) ≠ f(a').
Therefore f is injective.
Proof D:
Let a, a'∈ R and suppose f(a) = f(a').
Then 7 − 4a = 7 − 4a'.
Subtract 7 from both sides, so −4a = −4a' .
Divide both sides by −4, so a = a'.
Thus f is injective.
Proof E:
Let a, a' ∈ R and suppose f(a) ≠ f(a').
Then 7 − 4a ≠ 7 − 4a'
and
−4a ≠ −4a'
and so
a ≠ a'
Thus f is injective.
Page 4 of 8
#10. Define f: R → R defined by f(x) = x2. Let S = [0, 9] and T = [− 9, 0].
#10(a) Find f(S), f(T), and f(S ∩ T). Is f(S ∩ T) = f(S) ∩ f(T)?
#10(b) Find f
−1
(S), f
−1
(T), and f
−1
(S ∩ T). Is f
−1
(S ∩ T) = f
−1
(S) ∩ f
−1
(T)?
#11. Classify each function as injective, surjective, bijective, or none of these, as appropriate. If not
injective, briefly explain. If not surjective, briefly explain. (Otherwise, no explanation required.)
#11 (a) f: Z → Z defined by f(n) = 5 − n
#11 (b) f: N → Z defined by f(n) = n2 + 2n
#11 (c) f: Z → Z defined by f(n) = n2 + 2n
Page 5 of 8
Cardinality
#12. For each statement, is it True or False? (no explanation required)
________ (a) ⋂ 6 − , 6 + is a countable set.
________ (b) ⋃ 6 − , 6 + is a countable set.
________ (c) No subset of the irrational numbers is countable.
________ (d) Every subset of the rational numbers is countable.
#13. State a specific function f that is a bijection f: (0, 1) → (1, ∞). (Note that you can then conclude that
intervals (0, 1) and (1,∞) have the same cardinality. You are not asked to carry out a formal proof that
your f is bijective.)
#14. HINT: Look at page 2 of my posted notes on Cardinality.
#14(a) Show that the intervals (0, 1) and (3, 8) have the same cardinality by finding a
bijection f:(0, 1) → (3, 8).
#14(b) Suppose r < s.
Prove that the intervals (0, 1) and (r, s) have the same cardinality by finding a bijection f:(0, 1) → (r, s).
#14(c) Suppose a < b and c < d. Our goal is to show that open intervals (a, b) and (c, d) have the same
cardinality. By part (b), there exist bijections g:(0, 1) → (a, b) and h:(0, 1) → (c, d).
State a specific function f that is a bijection f: (a, b) → (c, d), where f is an appropriate composition of
functions involving functions g, h, and/or their inverses. [Recall composition of functions ---see Relations and
Functions notes, page 9]. You do not need a complicated formula. Just make use some of the functions g,
h, g−1, h−1, with an appropriate composition.
Page 6 of 8
ORDERED FIELDS
#15. Consult the list of properties A1 - A5, M1 - M5, DL, O1 - O4 from my Ordered Field notes.
Rather than considering those properties applied on the set R of real numbers, restrict the set as
indicated below. In other words, check which properties still apply, when R is replaced by the set
specified.
#15 (a) Which properties are not satisfied on the set N? (Just list the identifying labels.)
#15(b) Which properties are not satisfied on the set Z? (Just list the identifying labels.)
#15(c) Let S = {x in R such that x ≥ 0}. Which properties are not satisfied on the set S? (Just list the
identifying labels.)
#16. Prove that 0 < 1/2 < 1. Fill in the blanks of the proof. Refer to the field axioms and order axioms
and the Theorem in my Ordered Field notes, pages 1-2.
Proof:
Note that 0 < 1 (by Theorem part ___)
Adding 1 to both sides, 0 + 1 < 1 + 1 by order axiom ___.
0 + 1 = 1 by field axiom _____, and 1 + 1 = successor of 1, which is designated by 2 (Peano axiom in
Induction notes).
So, we have 1 < 2.
Since 0 < 1 and 1 < 2, we have 0 < 1 < 2.
Then 0 < 1/2 < 1/1 by Theorem, part ___
Note that 1/1 = 1 (because 1 ⋅ 1 = 1).
Thus, we have 0 < 1/2 < 1, as desired.
Page 7 of 8
#17. Let x be a real number.
Claim: If 0 ≤ x < ε for all real numbers ε > 0, then x = 0.
Fill in the blanks of the proof of the claim. Refer to the field axioms and order axioms and the Theorem
in my Ordered Field notes, pages 1-2.
Proof of Claim, by contradiction:
Suppose not.
Suppose for all real numbers ε > 0, we have 0 ≤ x < ε , but x ≠ 0.
Since 0 ≤ x and x ≠ 0, we must have x __ 0. (fill in blank with , whichever is appropriate)
Set ε = (1/2) x.
Since 1/2 > 0 (by problem #16) ,
we have ε = (1/2) x > (1/2) ⋅ 0 by order axiom ____.
=0
since (1/2) ⋅ 0 = 0 by Theorem, part ___.
Since 1/2 < 1 (by problem #16) and x __ 0, we have ε = (1/2) x < 1 ⋅ x by order axiom ____,
and so ε = (1/2) x < x
since 1 ⋅ x = x by field axiom ____.
In summary, we know x __ 0 and we have found a particular ε > 0 for which x __ ε. (fill in the blanks to
show the correct order relationships between the numbers)
We have reached a contradiction of our hypothesis that x < ε, and so we conclude that x must be equal
to 0.
Page 8 of 8
Ordered Fields
Assume the existence of a set R, called the set of real numbers, and two operations + and ⋅, called
addition and multiplication, such that the following properties apply:
Field Axioms
A1. Closure property of addition: For all x, y ∈ R, x + y ∈ R and if x = w and y = z, then x + y = w + z.
A2. Commutative law of addition: For all x, y ∈ R, x + y = y + x.
A3. Associative law of addition: For all x, y, z ∈ R, x + (y + z) = (x + y) + z.
A4. Identity for addition: There is a unique real number 0 such that x + 0 = x, for all x ∈ R.
A5. Additive inverse: For each x ∈ R, there is a unique real number -x such that x + (-x) = 0.
M1. Closure property of multiplication: For all x, y, ∈ R, x ⋅ y ∈ R, and if x = w and y = z, then x ⋅ y = w ⋅ z.
M2. Commutative law of multiplication: For all x, y ∈ R, x ⋅ y = y ⋅ x.
M3. Associative law of multiplication: For all x, y, z ∈ R, x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z.
M4. Identity for multiplication:
There is a unique real number 1 such that 1 is not 0 and x ⋅ 1 = x for all x ∈ R.
M5. Multiplicative inverse: For each x ∈ R with x ⋅ 0, there is a unique real number 1/x
such that x ⋅ (1/x) = 1. We also write x-1 in place of 1/x.
DL. Distributive law: For all x, y, z ∈ R, x ⋅ (y + z) = x ⋅ y + x ⋅ z.
Order axioms (shown below for )
O1. Trichotomy law: For all x, y ∈ R, exactly one of the relations x = y, x > y, or x < y holds.
O2. Transitive property: For all x, y, z ∈ R, if x < y and y < z, then x < z.
O3. Addition principle for inequalities: For all x, y, z ∈ R, if x < y, then x + z < y + z.
O4. Multiplication principle for inequalities: For all x, y, z ∈ R, if x < y and z > 0, then x ⋅ z < y ⋅ z.
An ordered field is any mathematical system satisfying the 15 axioms.
Discussed in the Lebl book in section 1.1
Examples: Q and R.
Page 1 of 2
Here is a sampling of results that can be proven.
Theorem. Let x, y, and z be real numbers.
(a) If x + z = y + z, then x = y.
(b) x ⋅ 0 = 0.
(c) (-1) ⋅ x = -x.
(d) x ⋅ y = 0 iff x = 0 or y = 0.
(e) x < y iff -y < -x.
(f)
If x < y and z < 0, then x ⋅ z > y ⋅ z.
(g) If x ≠ 0 , then x2 > 0.
(h) 0 < 1.
(i) If 0 < x < y, then 0 < 1/y < 1/x.
Proof of (e):
Suppose that
x < y.
x + (-x) < y + (-x) by O3 (Addition principle for inequalities)
0 < [y + (-x)] by A5 (Additive inverse)
0 + (-y) < [y + (-x)] + (-y) by O3.
0 + (-y) < y +[ (-x)] + (-y)] by A3 (Associative property for +).
(-y) + 0 < y +[ (-y)] + (-x)] by A2 (Commutative property for +).
-y < [y + (-y)] + (-x) by A4 (Identity for +) and A3 (Associative property for +)
-y < 0 + (-x)
by A5 (Additive inverse)
-y < (-x) + 0
by A2 (Commutative property for +).
-y < -x
by A4 (Identity for +)
So, if x < y, then -y < -x.
Suppose that -y < -x.
Then -(-x) < -(-y) (applying a < b ⇒ -b< -a, with a = -y and b = -x)
x
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