The relation RR in the natural numbers N+={1,2,3,…}N+={1,2,3,…} is defined by ==.
So, R⊂N+×N+R⊂N+×N+ and R={⟨x,y⟩:x,y∈N+,x=y}R={⟨x,y⟩:x,y∈N+,x=y}.

Irreflexive:   ∀x : ⟨x,x⟩∉R  ∀x : ⟨x,x⟩∉R

Asymmetric:   ∀x,y : ⟨x,y⟩∈R → ⟨y,x⟩∉R  ∀x,y : ⟨x,y⟩∈R → ⟨y,x⟩∉R

Anti-symmetric:   ∀x,y : (⟨x,y⟩∈R ∧ ⟨y,x⟩∈R) → x=y  ∀x,y : (⟨x,y⟩∈R ∧ ⟨y,x⟩∈R) → x=y

Intransitive:   ∀x,y,z : (⟨x,y⟩∈R ∧ ⟨y,z⟩∈R) → ⟨x,z⟩∉R

Let's break down the given properties one by one to determine whether the relation \( R \) defined as \( R = \{ \langle x, y \rangle : x, y \in \mathbb{N}^+, x = y \} \) satisfies them.

### 1. **Irreflexive:**
- A relation \( R \) is irreflexive if \( \forall x : \langle x, x \rangle \notin R \).
- However, by the definition of \( R \), \( \langle x, x \rangle \in R \) for all \( x \in \mathbb{N}^+ \) since the relation only contains pairs where \( x = y \).
- **Conclusion:** The relation is **not irreflexive** because \( \langle x, x \rangle \in R \) for all \( x \).

### 2. **Asymmetric:**
- A relation \( R \) is asymmetric if \( \forall x, y : \langle x, y \rangle \in R \rightarrow \langle y, x \rangle \notin R \).
- In the case of \( R \), we only have pairs where \( x = y \), meaning \( \langle x, y \rangle = \langle y, x \rangle \), so symmetry is always present when \( x = y \).
- **Conclusion:** The relation is **not asymmetric** because \( \langle x, y \rangle \in R \) implies \( \langle y, x \rangle \in R \) when \( x = y \).

### 3. **Anti-symmetric:**
- A relation \( R \) is anti-symmetric if \( \forall x, y : (\langle x, y \rangle \in R \wedge \langle y, x \rangle \in R) \rightarrow x = y \).
- In \( R \), the only pairs that exist are \( \langle x, x \rangle \), which trivially satisfies the anti-symmetry condition because if \( \langle x, y \rangle \in R \) and \( \langle y, x \rangle \in R \), then \( x = y \).
- **Conclusion:** The relation **is anti-symmetric**.

### 4. **Intransitive:**
- A relation \( R \) is intransitive if \( \forall x, y, z : (\langle x, y \rangle \in R \wedge \langle y, z \rangle \in R) \rightarrow \langle x, z \rangle \notin R \).
- Since \( R \) only includes pairs where \( x = y \), this condition doesn't hold because if \( \langle x, y \rangle \in R \) and \( \langle y, z \rangle \in R \), then \( x = y = z \), and thus \( \langle x, z \rangle \in R \).
- **Conclusion:** The relation is **not intransitive** because the transitivity condition holds.

### Final Summary:
- The relation \( R \) is:
  - **Not irreflexive**
  - **Not asymmetric**
  - **Anti-symmetric**
  - **Not intransitive**

Would you like more details or clarifications?

Here are 5 related questions to expand on the topic:
1. What is the difference between asymmetric and anti-symmetric relations?
2. Can a relation be both reflexive and anti-symmetric?
3. How do transitive and intransitive relations differ in mathematical terms?
4. What are examples of irreflexive relations in real-world applications?
5. Can a relation be symmetric and anti-symmetric at the same time?

**Tip:** Anti-symmetric relations allow pairs \( \langle x, y \rangle \) and \( \langle y, x \rangle \) only when \( x = y \), so reflexive relations can still be anti-symmetric.

The relation R in the natural numbers N+={1,2,3,…} is defined by x=y. Is the relation R irreflexive, asymmetric, anti-symmetric, and intransitive?

This problem explores the properties of relations in set theory, focusing on reflexivity, asymmetry, anti-symmetry, and transitivity. Learn whether the relation R, defined as pairs of natural numbers (x=y), is irreflexive, asymmetric, anti-symmetric, and intransitive. Suitable for undergraduate math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation