Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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For each of the following relations, determine whether the relation is:
• Reflexive.
• Anti-reflexive.
• Symmetric.
• Anti-symmetric.
• Transitive.
• A partial order.
• A strict order.
• An equivalence relation.
Justify all your answers.
a. ? is a relation on the set of all people such that (?, ?) ∈ ? if and only if ? and ? have a common grandparent.
b. ? is a relation on the power set of a set ? such that (?, ?) ∈ ? if and only if ? ⊂ ?.
c. ? is a relation on ℤ such that (?, ?) ∈ ? if and only if ?? ≥ ?.
d. ? is a relation on ℤ + such that (?, ?) ∈ ? if and only if there is a positive integer ? such that ?^? = ?.
e. ? is a relation on ℝ such that (?, ?) ∈ ? if and only if ? − ? is rational. Hint: The sum of two rational numbers is rational.
Transcribed Image Text:**Task:**
For each of the following relations, determine whether the relation is:
- Reflexive.
- Anti-reflexive.
- Symmetric.
- Anti-symmetric.
- Transitive.
- A partial order.
- A strict order.
- An equivalence relation.
*Justify all your answers.*
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**a.** \( R \) is a relation on the set of all people such that \((a, b) \in R\) if and only if \(a\) and \(b\) have a common grandparent.
**b.** \( R \) is a relation on the power set of a set \( A \) such that \((X, Y) \in R\) if and only if \(X \subseteq Y\).
**c.** \( R \) is a relation on \( \mathbb{Z} \) such that \((x, y) \in R\) if and only if \(xy \geq 0\).
**d.** \( R \) is a relation on \( \mathbb{Z}^+ \) such that \((x, y) \in R\) if and only if there is a positive integer \( n \) such that \( x^n = y \).
**e.** \( R \) is a relation on \( \mathbb{R} \) such that \((x, y) \in R\) if and only if \(x - y\) is rational. *(Hint: The sum of two rational numbers is rational.)*
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