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11. Suppose A = {a,b,c,d} and R = {(a, a),(b,b), (c,c), (d,d)}. Is R reflexive? Symmet- ric? Transitive? If a property does not hold, say why. 12. Prove that the relation | (divides) on the set Z is reflexive and transitive. (Use Example 16.8 as a guide if you are unsure of how to proceed.) 13. Consider the relation R = {(x, y) eRxR:x-ye Z} on R. Prove that this relation is reflexive, symmetric and transitive.

11. Suppose A = {a,b,c,d} and R = {(a, a),(b,b), (c,c), (d,d)}. Is R reflexive? Symmet- ric? Transitive? If a property does not hold, say why. 12. Prove that the relation | (divides) on the set Z is reflexive and transitive. (Use Example 16.8 as a guide if you are unsure of how to proceed.) 13. Consider the relation R = {(x, y) eRxR:x-ye Z} on R. Prove that this relation is reflexive, symmetric and transitive.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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--- **Exercise 11:** Suppose \( A = \{ a, b, c, d \} \) and \( R = \{ (a, a), (b, b), (c, c), (d, d) \} \). - **Question:** Is \( R \) reflexive, symmetric, and transitive? If a property does not hold, provide a justification. **Exercise 12:** Prove that the relation \( | \) (divides) on the set \( \mathbb{Z} \) is reflexive and transitive. - **Note:** Use Example 16.8 as a guide if you are unsure of how to proceed. **Exercise 13:** Consider the relation \( R = \{ (x, y) \in \mathbb{R} \times \mathbb{R} : x - y \in \mathbb{Z} \} \) on \( \mathbb{R} \). - **Question:** Prove that this relation is reflexive, symmetric, and transitive. --- Exercises revolve around understanding the properties of relations such as reflexivity, symmetry, and transitivity within given sets and conditions.
Transcribed Image Text:--- **Exercise 11:** Suppose \( A = \{ a, b, c, d \} \) and \( R = \{ (a, a), (b, b), (c, c), (d, d) \} \). - **Question:** Is \( R \) reflexive, symmetric, and transitive? If a property does not hold, provide a justification. **Exercise 12:** Prove that the relation \( | \) (divides) on the set \( \mathbb{Z} \) is reflexive and transitive. - **Note:** Use Example 16.8 as a guide if you are unsure of how to proceed. **Exercise 13:** Consider the relation \( R = \{ (x, y) \in \mathbb{R} \times \mathbb{R} : x - y \in \mathbb{Z} \} \) on \( \mathbb{R} \). - **Question:** Prove that this relation is reflexive, symmetric, and transitive. --- Exercises revolve around understanding the properties of relations such as reflexivity, symmetry, and transitivity within given sets and conditions.
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