MAT 299 Module Five

MAT 299 Module Five Homework General:  Before beginning this homework, be sure to read the textbook sections and the material in Module Five.  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 except for the symbol 𝒢 or “script G” which is in Cambria math (it is not a basic Unicode symbol).  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. SNHU MAT299 Page 1 of 3 Module Five Homework 1. Suppose ℱ and 𝒢 are families of sets. Prove that ∪ℱ and ∪𝒢 are not disjoint iff there exists A ∈ ℱ and B ∈ 𝒢 where A and B are not disjoint. This problem is similar to examples and exercises in Section 3.4 of your SNHU MAT299 textbook. 2. Prove that for every integer n, 30 | n iff 5 | n and 6 | n. This problem is similar to examples and exercises in Section 3.4 of your SNHU MAT299 textbook. 3. Prove that there is a unique real number x such that for every real number y, xy + x – 17 = 17y. This problem is similar to examples and exercises in Section 3.6 of your SNHU MAT299 textbook. 4. Let U be any set. Prove that for every B ∈ ℘(U) there is a unique D ∈ ℘(U) such that for every C ∈ ℘(U), C \ B = C ∩ D. This problem is similar to examples and exercises in Section 3.6 of your SNHU MAT299 textbook. 5. For every positive integer n, there is a sequence of 2n consecutive positive integers containing no primes. Either provide a proof to show that this is true or provide a counterexample to show that this is false. This problem is similar to examples and exercises in Section 3.7 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 3 Module Five Homework 6. Consider the function f(x) = x2 for 0 ≤ x ≤ 10. Prove that limx→5 f(x) = 25. Note that proofs of limits involve the epsilon / delta (or ε / δ) method. Hint: What is the maximum value of x + 5 on the interval [0, 10]? This problem is similar to examples and exercises in Section 3.7 of your SNHU MAT299 textbook. SNHU MAT299 Page 3 of 3 Module Five Homework MAT 299 Module Five Homework General: • Before beginning this homework, be sure to read the textbook sections and the material in Module Five. • 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 except for the symbol 𝒢 or “script G” which is in Cambria math (it is not a basic Unicode symbol). • 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. SNHU MAT299 Page 1 of 3 Module Five Homework 1. Suppose ℱ and 𝒢 are families of sets. Prove that ∪ℱ and ∪𝒢 are not disjoint iff there exists A ∈ ℱ and B ∈ 𝒢 where A and B are not disjoint. This problem is similar to examples and exercises in Section 3.4 of your SNHU MAT299 textbook. 2. Prove that for every integer n, 30 | n iff 5 | n and 6 | n. This problem is similar to examples and exercises in Section 3.4 of your SNHU MAT299 textbook. 3. Prove that there is a unique real number x such that for every real number y, xy + x – 17 = 17y. This problem is similar to examples and exercises in Section 3.6 of your SNHU MAT299 textbook. 4. Let U be any set. Prove that for every B ∈ ℘(U) there is a unique D ∈ ℘(U) such that for every C ∈ ℘(U), C \ B = C ∩ D. This problem is similar to examples and exercises in Section 3.6 of your SNHU MAT299 textbook. 5. For every positive integer n, there is a sequence of 2n consecutive positive integers containing no primes. Either provide a proof to show that this is true or provide a counterexample to show that this is false. This problem is similar to examples and exercises in Section 3.7 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 3 Module Five Homework 6. Consider the function f(x) = x2 for 0 ≤ x ≤ 10. Prove that limx→5 f(x) = 25. Note that proofs of limits involve the epsilon / delta (or ε / δ) method. Hint: What is the maximum value of x + 5 on the interval [0, 10]? This problem is similar to examples and exercises in Section 3.7 of your SNHU MAT299 textbook. SNHU MAT299 Page 3 of 3 Module Five Homework
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