give reason/ example (one of finite set, one of infinite set) 
are all transitive relations anti symmetric?

### Transitive and Anti-symmetric Relations

To address the question of whether all transitive relations are anti-symmetric, let’s break it down:

1. **Transitive Relation**: A relation \( R \) on a set \( A \) is called transitive if, for all \( a, b, c \in A \), whenever \( aRb \) and \( bRc \), then \( aRc \) holds.

2. **Anti-symmetric Relation**: A relation \( R \) on a set \( A \) is called anti-symmetric if, for all \( a, b \in A \), whenever \( aRb \) and \( bRa \), then \( a = b \).

Now, to explore whether all transitive relations are also anti-symmetric, let’s look at examples for both finite and infinite sets.

### 1. **Finite Set Example (Relation is Transitive but NOT Anti-symmetric)**

Consider the set \( A = \{1, 2, 3\} \) and the relation \( R = \{(1, 2), (2, 3), (1, 3), (2, 2)\} \) on \( A \). 
- **Transitivity**: The relation is transitive because:
  - \( (1, 2) \) and \( (2, 3) \) imply \( (1, 3) \), which is included in the relation.
  - \( (2, 2) \) and \( (2, 2) \) imply \( (2, 2) \), which is already there.
- **Anti-symmetry**: This relation is **not anti-symmetric** because \( (1, 2) \) and \( (2, 1) \) are not both present, but if we added both, it would violate anti-symmetry as \( 1 \neq 2 \).

Thus, this is a counterexample of a transitive relation that is not anti-symmetric on a finite set.

### 2. **Infinite Set Example (Relation is Transitive and Anti-symmetric)**

Consider the set \( A = \mathbb{N} \) (the set of natural numbers) and the relation \( R = \leq \) (the "less than or equal to" relation) on \( \mathbb{N} \).
- **Transitivity**: If \( a \leq b \) and \( b \leq c \), then \( a \leq c \). This satisfies transitivity.
- **Anti-symmetry**: If \( a \leq b \) and \( b \leq a \), then \( a = b \). This satisfies anti-symmetry.

Thus, this is an example of a relation that is both transitive and anti-symmetric on an infinite set.

### Conclusion:
Not all transitive relations are anti-symmetric. The relation on finite sets may be transitive without being anti-symmetric, as shown in the finite example. However, some relations, such as the "less than or equal to" relation on an infinite set, can be both transitive and anti-symmetric.

### Would you like further clarification on this, or have any additional questions?

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Here are 5 related questions to consider:
1. Can a relation be anti-symmetric but not transitive?
2. How does symmetry interact with transitivity and anti-symmetry?
3. What are real-world examples of transitive but not anti-symmetric relations?
4. Is the equality relation both transitive and anti-symmetric?
5. How does reflexivity affect the properties of relations like transitivity and anti-symmetry?

**Tip**: When examining relations, always test their properties individually with concrete examples to avoid confusion!

give reason/ example (one of finite set, one of infinite set) are all transitive relations anti symmetric?

Explore whether all transitive relations are anti-symmetric through examples of finite and infinite sets. This problem highlights key concepts from set theory and relational properties, including transitivity and anti-symmetry. Suitable for students in Grades 10-12.

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