MATH 301 Week 1 Homework, Spring, 2016. Due Monday, February 7.
NAME: _____________________________________
Instructions: You are encouraged to discuss homework, consult (but not plagiarize) 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.
Throughout, N refers to the set of natural numbers, {1, 2, 3, ...} , Z refers to the set of integers, and R refers to the
set of real numbers. The symbol "∋" means "such that". The symbol "∈" means "is an element of" or "in". Often, I
just write "in" rather than "∈" because "in" is so much easier and quicker to type.
Basic Set Operations
#1. Let A = {2, 5, 6, 9}, B = {1, 2, 3, 4, 5}, and C = {2, 3, 7, 8}.
Find each of the following sets. Use set notation with the elements listed { _, _, _, ... }.
(a) A ∩ B
(b) B ∪ C
(c) (A ∩ B) \ C
(d) C \ (A ∩ B)
#2. Consider the intervals [5, 8), (3, 6], and (4, 7] of real numbers.
(No work required to be shown).
(a) State the union of these three intervals:
(b) State the intersection of these three intervals:
Remarks: Write your answers in interval notation, as simply as possible. To find the answers, it can be
helpful to graph the intervals on a number line (no requirement to submit graph).
#3. Consider the collection of intervals of real numbers = − , 1 − , where n ∈ N.
(No work required to be shown).
(a) State the intervals corresponding to A1, A2, and A3.
(b) State the union of the whole collection of intervals ⋃
(c) State the intersection of the whole collection of intervals ⋂
Page 1 of 8
Basic Logic
#4. TRUE/FALSE. Determine the truth value of each sentence (no explanation required).
________(a) A statement is a sentence that is true.
________(b) If ~p is true, then p is false.
________(c) The phrase "not p or not q" means " not both p and q".
________(d) In logic, p ∨ q refers to the "inclusive or" and is true for "either p or q or both."
________(e) The conditional statement p ⇒ q is true if and only if p and q are true.
________(f) The negation of p ⇒ ~q is p ⇒ q.
#5. Write the negation of each of the following statements in a clearly worded English sentence.
(a) If Jennifer is late, then Jennifer misses the bus.
(b) Tom is reading a book and Tom is drinking coffee.
#6. Construct a truth table for the statement [~q ∧ ( p ⇒ q )] ⇒ ~p
Show intermediate steps, with appropriate column headings.
p
q
T
T
T
F
F
T
F
F
Page 2 of 8
Quantifiers
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required).
________(a) ∃ n in N ∋ √ = 3.
________(b) ∀ x in R, x2 + 4 ≠ 0.
________(c) ∀m, n in N, m − n is in N.
________(d) ∀ x in R, if |x + 1| > 4, then x > 3.
#8. For each statement,
(i) write the statement in logical form with appropriate variables and quantifiers,
(ii) write the negation in logical form,
and (iii) write the negation in a clearly worded English sentence.
(a) Some insects can fly.
(b) All real numbers are positive.
(c) No woman has served as U.S. President.
Page 3 of 8
Quantifiers , Counterexamples, Disproof
#9. For each statement, decide if the statement is true or false. If false, explain; provide a
counterexample as appropriate or a careful explanation. (If true, no explanation expected)
(a) ∀ n in N,
≥
(b) ∀ nonzero x in R ,
≥ .
(c) ∀ n in N, 14 + 3n is prime.
(d) ∃ k in N ∋ ∀ n in N, n ≥ k.
(e) ∀ x, y in R, if x > 0 and y > 0, then x + y ≤ xy.
(f) ∀x in R, ∃ n in N ∋ n ≥ x .
(g) ∃ n in N ∋ ∀x in R, n ≥ x .
Page 4 of 8
Applications of logic; Proofs
#10. Write the converse, the inverse, and the contrapositive of the statement
If Amy is a voter, then Amy is at least 18 years old.
Definitions: A real number r is rational iff ∃ integers m and n such that r = m/n and n is nonzero.
A real number s is irrational iff s is not rational.
#11. Consider the following statement:
For all real numbers x and y, if the product xy is irrational, then x is irrational and y is irrational.
(a) Carefully state the contrapositive.
(b) Is the contrapositive true or false? Explain.
#12. A father keeps a set of spare keys at home. His family needed one of the spare keys while the
father was traveling and out of communication. The father kept a set of notes about the location of the
keys (in fact, more clues than needed). Here are the five clues:
If the family cat is named Cassie, then the keys are in the shed.
If the home has a pool, then the keys are not in the shed.
If the keys are in the desk, then the family cat is not named Cassie.
The home has a pool.
The family cat is named Cassie or the keys are in the freezer.
Let L, C, K, G, and B represent the following statements.
C = “The family cat is named Cassie.”
P = “The home has a pool.”
S = “The keys are in the shed.”
D = “The keys are in the desk.”
F = “The keys are in the freezer.”
(a) Write the five clues as five statements in symbolic form, using symbols ⇒, ∨, ∧, ~ as needed.
[Do not overcomplicate matters -- there are no quantifiers involved in this problem.]
(b) Using the clues, what do you conclude? Are the keys in the shed, desk, or freezer? EXPLAIN carefully
how you arrived at your conclusion. (space provided on next page)
Page 5 of 8
Recall the definitions of even, odd, and multiple of a. (These are used in #13 and #14.)
An integer n is even iff n = 2k for some integer k.
An integer n is odd iff n = 2k + 1 for some integer k.
An integer n is a multiple of a iff n = ak for some integer k. (When a = 2, this is exactly the definition of even.)
#13. Prove carefully: For any integers p and q, if p is odd and q is even, then 4p + q − 3 is odd.
Page 6 of 8
#14. Claim: For all integers p and q, if their sum p + q is even, then p and q are even.
Consider the following "proofs" of the claim.
Proof A:
Suppose p and q are any even integers. By definition of even, ∃ integer k such that p = 2k and q = 2k.
Then p + q = 2k + 2k = 2(2k). Let m = 2k, which is an integer.
Thus, p + q = 2m for some integer m, and by definition of even, p + q is even.
Proof B:
Suppose p and q are any even integers.
By definition of even, ∃ integer k such that p = 2k and ∃ integer n such that q = 2n.
Then p + q = 2k + 2n = 2(k + n). Let m = k + n, which is an integer.
Thus, p + q = 2m for some integer m, and by definition of even, p + q is even.
Proof C:
(By contraposition; i.e., proving the contrapositive)
Suppose p or q is odd. We want to show that p + q is odd.
Suppose it is p that's odd, with q even. (A similar argument applies for p even, with q odd; just switch roles of p & q.)
Then ∃ integer m such that p = 2m + 1 and ∃ integer n such that q = 2n.
Then p + q = (2m + 1) + 2n = 2(m + n) + 1.
Let k = 2m + n, which is an integer.
So, p + q = 2k + 1 for some integer k, and by definition of odd, p + q is odd.
Proof D:
Suppose p and q are integers and p + q is even.
By definition of even, ∃ integer k such that p + q = 2k = 2(m + n) = 2m + 2n for some integers m and n.
Then p = 2m and q = 2n for integers m and n.
By definition of even, p and q are even integers.
INSTRUCTIONS:
(a) Critique each proof (A, B, C, D). For each proof, is it a logically valid argument proving the claim?
What are the flaws, if any?
(b) Is the Claim true or false? Explain.
More space provided on next page
Page 7 of 8
#15. Fill in the blanks to complete the proof of the following statement:
For all sets A, B, and C, if A ∩ C = φ and B ⊆ C, then A ∩ B = φ.
Proof (by contradiction):
Let A, B, and C be any sets.
Suppose ________________ and ______________ but A ∩ B ≠ φ.
Since A ∩ B ≠ φ, there exists x ∈ ___________.
Since x ∈ A ∩ B, x ∈ ________ and x ∈ _______.
Since x ∈ B and B ⊆ C, we have x ∈ ______.
Thus x ∈ A and x ∈ C, so x ∈ ____________.
But this contradicts our hypothesis that ____________.
So, we conclude that A and B cannot have any elements in common, that A ∩ B = φ.
Page 8 of 8
MATH 301 Week 1 Homework, Spring, 2016. Due Monday, February 7.
NAME: _____________________________________
Instructions: You are encouraged to discuss homework, consult (but not plagiarize) 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.
Throughout, N refers to the set of natural numbers, {1, 2, 3, ...} , Z refers to the set of integers, and R refers to the
set of real numbers. The symbol "" means "such that". The symbol "" means "is an element of" or "in". Often, I
just write "in" rather than "" because "in" is so much easier and quicker to type.
Basic Set Operations
#1. Let A = {2, 5, 6, 9}, B = {1, 2, 3, 4, 5}, and C = {2, 3, 7, 8}.
Find each of the following sets. Use set notation with the elements listed { _, _, _, ... }.
(a) A B
(b) B C
(c) (A B) \ C
(d) C \ (A B)
#2. Consider the intervals [5, 8), (3, 6], and (4, 7] of real numbers.
(No work required to be shown).
(a) State the union of these three intervals:
(b) State the intersection of these three intervals:
Remarks: Write your answers in interval notation, as simply as possible. To find the answers, it can be
helpful to graph the intervals on a number line (no requirement to submit graph).
1
1
#3. Consider the collection of intervals of real numbers 𝐴𝑛 = [− 𝑛 , 1 − 𝑛], where n N.
(No work required to be shown).
(a) State the intervals corresponding to A1, A2, and A3.
(b) State the union of the whole collection of intervals ⋃∞
𝑛 = 1 𝐴𝑛
(c) State the intersection of the whole collection of intervals ⋂∞
𝑛 = 1 𝐴𝑛
Page 1 of 8
Basic Logic
#4. TRUE/FALSE. Determine the truth value of each sentence (no explanation required).
________(a) A statement is a sentence that is true.
________(b) If ~p is true, then p is false.
________(c) The phrase "not p or not q" means " not both p and q".
________(d) In logic, p q refers to the "inclusive or" and is true for "either p or q or both."
________(e) The conditional statement p q is true if and only if p and q are true.
________(f) The negation of p ~q is p q.
#5. Write the negation of each of the following statements in a clearly worded English sentence.
(a) If Jennifer is late, then Jennifer misses the bus.
(b) Tom is reading a book and Tom is drinking coffee.
#6. Construct a truth table for the statement [~q ( p q )] ~p
Show intermediate steps, with appropriate column headings.
p
q
T
T
T
F
F
T
F
F
Page 2 of 8
Quantifiers
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required).
________(a) n in N √𝑛 = 3.
________(b) x in R, x2 + 4 0.
________(c) m, n in N, m − n is in N.
________(d) x in R, if |x + 1| > 4, then x > 3.
#8. For each statement,
(i) write the statement in logical form with appropriate variables and quantifiers,
(ii) write the negation in logical form,
and (iii) write the negation in a clearly worded English sentence.
(a) Some insects can fly.
(b) All real numbers are positive.
(c) No woman has served as U.S. President.
Page 3 of 8
Quantifiers , Counterexamples, Disproof
#9. For each statement, decide if the statement is true or false. If false, explain; provide a
counterexample as appropriate or a careful explanation. (If true, no explanation expected)
(a) n in N,
𝑛2 + 1
𝑛
≥𝑛
(b) nonzero x in R ,
𝑥2+ 1
𝑥
≥ 𝑥.
(c) n in N, 14 + 3n is prime.
(d) k in N n in N, n k.
(e) x, y in R, if x > 0 and y > 0, then x + y xy.
(f) x in R, n in N n x .
(g) n in N x in R, n x .
Page 4 of 8
Applications of logic; Proofs
#10. Write the converse, the inverse, and the contrapositive of the statement
If Amy is a voter, then Amy is at least 18 years old.
Definitions: A real number r is rational iff integers m and n such that r = m/n and n is nonzero.
A real number s is irrational iff s is not rational.
#11. Consider the following statement:
For all real numbers x and y, if the product xy is irrational, then x is irrational and y is irrational.
(a) Carefully state the contrapositive.
(b) Is the contrapositive true or false? Explain.
#12. A father keeps a set of spare keys at home. His family needed one of the spare keys while the
father was traveling and out of communication. The father kept a set of notes about the location of the
keys (in fact, more clues than needed). Here are the five clues:
If the family cat is named Cassie, then the keys are in the shed.
If the home has a pool, then the keys are not in the shed.
If the keys are in the desk, then the family cat is not named Cassie.
The home has a pool.
The family cat is named Cassie or the keys are in the freezer.
Let L, C, K, G, and B represent the following statements.
C = “The family cat is named Cassie.”
P = “The home has a pool.”
S = “The keys are in the shed.”
D = “The keys are in the desk.”
F = “The keys are in the freezer.”
(a) Write the five clues as five statements in symbolic form, using symbols , , , ~ as needed.
[Do not overcomplicate matters -- there are no quantifiers involved in this problem.]
(b) Using the clues, what do you conclude? Are the keys in the shed, desk, or freezer? EXPLAIN carefully
how you arrived at your conclusion. (space provided on next page)
Page 5 of 8
Recall the definitions of even, odd, and multiple of a. (These are used in #13 and #14.)
An integer n is even iff n = 2k for some integer k.
An integer n is odd iff n = 2k + 1 for some integer k.
An integer n is a multiple of a iff n = ak for some integer k. (When a = 2, this is exactly the definition of even.)
#13. Prove carefully: For any integers p and q, if p is odd and q is even, then 4p + q − 3 is odd.
Page 6 of 8
#14. Claim: For all integers p and q, if their sum p + q is even, then p and q are even.
Consider the following "proofs" of the claim.
Proof A:
Suppose p and q are any even integers. By definition of even, integer k such that p = 2k and q = 2k.
Then p + q = 2k + 2k = 2(2k). Let m = 2k, which is an integer.
Thus, p + q = 2m for some integer m, and by definition of even, p + q is even.
Proof B:
Suppose p and q are any even integers.
By definition of even, integer k such that p = 2k and integer n such that q = 2n.
Then p + q = 2k + 2n = 2(k + n). Let m = k + n, which is an integer.
Thus, p + q = 2m for some integer m, and by definition of even, p + q is even.
Proof C:
(By contraposition; i.e., proving the contrapositive)
Suppose p or q is odd. We want to show that p + q is odd.
Suppose it is p that's odd, with q even. (A similar argument applies for p even, with q odd; just switch roles of p & q.)
Then integer m such that p = 2m + 1 and integer n such that q = 2n.
Then p + q = (2m + 1) + 2n = 2(m + n) + 1.
Let k = 2m + n, which is an integer.
So, p + q = 2k + 1 for some integer k, and by definition of odd, p + q is odd.
Proof D:
Suppose p and q are integers and p + q is even.
By definition of even, integer k such that p + q = 2k = 2(m + n) = 2m + 2n for some integers m and n.
Then p = 2m and q = 2n for integers m and n.
By definition of even, p and q are even integers.
INSTRUCTIONS:
(a) Critique each proof (A, B, C, D). For each proof, is it a logically valid argument proving the claim?
What are the flaws, if any?
(b) Is the Claim true or false? Explain.
More space provided on next page
Page 7 of 8
#15. Fill in the blanks to complete the proof of the following statement:
For all sets A, B, and C, if A C = and B C, then A B = .
Proof (by contradiction):
Let A, B, and C be any sets.
Suppose ________________ and ______________ but A B .
Since A B , there exists x ___________.
Since x A B, x ________ and x _______.
Since x B and B C, we have x ______.
Thus x A and x C, so x ____________.
But this contradicts our hypothesis that ____________.
So, we conclude that A and B cannot have any elements in common, that A B = .
Page 8 of 8
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