# 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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