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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