mathematical symbols

MAT 299 Module 6 Homework General:  Before beginning this homework, be sure to read the textbook sections and the material in Learning Module 6.  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.  These problems are proprietary to SNHU COCE, and they may not be posted on any non-SNHU web site.  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 6 Homework MAT 299 Module 6 Homework 1. Suppose that A is a set and {Bi | i ∈ I} is an indexed families of sets. Prove that A × (Ui∈I Bi) = Ui∈I (A × Bi). This problem is similar to examples and exercises in Section 4.1 of your SNHU MAT299 textbook. 2. Suppose that A = {1, 2, 3}, B = {4, 5}, C = {a, b, c, d}, R = {(1, b), (2, a), (2, b), (2, c), (3, d)} and S = {(4, a), (4, d), (5, b), (5, c)}. Note that R is a relation from A to C and S is a relation from B to C. a. Find S–1 ○ R. This is a relation from which set to which other set? Justify your solution. b. Find R–1 ○ S. This is a relation from which set to which other set? Justify your solution. This problem is similar to examples and exercises in Section 4.2 of your SNHU MAT299 textbook. 3. Suppose R and S are relations from A to B. Must the following statements be true? Justify your answers with proofs or counterexamples. a. R = Dom(R) × Ran(R) b. (R ∩ S)–1 = R–1 ∩ S–1 This problem is similar to examples and exercises in Section 4.2 of your SNHU MAT299 textbook. 4. List the ordered pairs in the relations represented by the following graph. Determine whether this relation is reflexive, symmetric, or transitive. Justify your answers with reasoning or counterexamples. This problem is similar to examples and exercises in Section 4.3 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 3 Module 6 Homework MAT 299 Module 6 Homework 5. Consider the function f: defined on positive integers with f(n) = n “flipped” as a mirror image into a decimal. For example, f(5) = .5, f(418) = .814, and f(1000) = .0001. Define a relation R on the positive integers as (m, n) ∈ R if and only if f(m) ≤ f(n). For example, (5, 418) ∈ R because .5 ≤ .814 but (418, .923) ∉ R because .814 > .329. Is R a partial order? 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 4.4 of your SNHU MAT299 textbook. 6. Define a relation R on ℤ as (a, b) ∈ R if and only if a and b, when written out, have the same number of 5s. For example, (1752, 95) ∈ R since they both have one 5 but (1752, 505) ∉ R since 1752 has one 5 but 505 has two 5s. Is R an equivalence relation? Prove that R is an equivalence relation. This problem is similar to examples and exercises in Section 4.5 of your SNHU MAT299 textbook. SNHU MAT299 Page 3 of 3 Module 6 Homework MAT 299 Module 6 Homework General: • Before beginning this homework, be sure to read the textbook sections and the material in Learning Module 6. • 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. • These problems are proprietary to SNHU COCE, and they may not be posted on any non-SNHU web site. • 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 6 Homework MAT 299 Module 6 Homework 1. Suppose that A is a set and {Bi | i ∈ I} is an indexed families of sets. Prove that A × (Ui∈I Bi) = Ui∈I (A × Bi). This problem is similar to examples and exercises in Section 4.1 of your SNHU MAT299 textbook. 2. Suppose that A = {1, 2, 3}, B = {4, 5}, C = {a, b, c, d}, R = {(1, b), (2, a), (2, b), (2, c), (3, d)} and S = {(4, a), (4, d), (5, b), (5, c)}. Note that R is a relation from A to C and S is a relation from B to C. a. Find S–1 ○ R. This is a relation from which set to which other set? Justify your solution. b. Find R–1 ○ S. This is a relation from which set to which other set? Justify your solution. This problem is similar to examples and exercises in Section 4.2 of your SNHU MAT299 textbook. 3. Suppose R and S are relations from A to B. Must the following statements be true? Justify your answers with proofs or counterexamples. a. R = Dom(R) × Ran(R) b. (R ∩ S)–1 = R–1 ∩ S–1 This problem is similar to examples and exercises in Section 4.2 of your SNHU MAT299 textbook. 4. List the ordered pairs in the relations represented by the following graph. Determine whether this relation is reflexive, symmetric, or transitive. Justify your answers with reasoning or counterexamples. This problem is similar to examples and exercises in Section 4.3 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 3 Module 6 Homework MAT 299 Module 6 Homework 5. Consider the function f: defined on positive integers with f(n) = n “flipped” as a mirror image into a decimal. For example, f(5) = .5, f(418) = .814, and f(1000) = .0001. Define a relation R on the positive integers as (m, n) ∈ R if and only if f(m) ≤ f(n). For example, (5, 418) ∈ R because .5 ≤ .814 but (418, .923) ∉ R because .814 > .329. Is R a partial order? 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 4.4 of your SNHU MAT299 textbook. 6. Define a relation R on ℤ as (a, b) ∈ R if and only if a and b, when written out, have the same number of 5s. For example, (1752, 95) ∈ R since they both have one 5 but (1752, 505) ∉ R since 1752 has one 5 but 505 has two 5s. Is R an equivalence relation? Prove that R is an equivalence relation. This problem is similar to examples and exercises in Section 4.5 of your SNHU MAT299 textbook. SNHU MAT299 Page 3 of 3 Module 6 Homework
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