Let R be a relation of set A of people. R={(a, b) : a is brother of b}. Is it an equivalence relation? Is
R partial ordering? (discard half brother, paternity brother)

Let's analyze the relation \( R \) on a set \( A \) of people, where \( R = \{(a, b) : a \text{ is the brother of } b \} \), under the assumption that we are only considering full brothers, and excluding half-brothers and paternity brothers.

### 1. **Equivalence Relation**

To determine if \( R \) is an equivalence relation, we need to check if it satisfies the following three properties:

#### a. Reflexive: 
A relation \( R \) is **reflexive** if \( (a, a) \in R \) for every \( a \in A \). This means that each person must be the brother of himself.

- However, no person is typically considered their own brother, so **reflexivity does not hold**.

#### b. Symmetric: 
A relation \( R \) is **symmetric** if \( (a, b) \in R \) implies \( (b, a) \in R \). This means if \( a \) is the brother of \( b \), then \( b \) must also be the brother of \( a \).

- If \( a \) is the brother of \( b \), it naturally follows that \( b \) is also the brother of \( a \). Therefore, **symmetry holds**.

#### c. Transitive: 
A relation \( R \) is **transitive** if \( (a, b) \in R \) and \( (b, c) \in R \) implies \( (a, c) \in R \). This means if \( a \) is the brother of \( b \), and \( b \) is the brother of \( c \), then \( a \) must also be the brother of \( c \).

- Transitivity does not hold in this case, because if \( a \) is the brother of \( b \) and \( b \) is the brother of \( c \), it does not necessarily mean that \( a \) is the brother of \( c \) (they could be cousins or unrelated in some other way). Hence, **transitivity does not hold**.

Since the relation fails to be reflexive and transitive, **\( R \) is not an equivalence relation**.

### 2. **Partial Ordering**

A relation is a **partial ordering** if it is:

#### a. Reflexive:
As mentioned above, the relation is not reflexive because no one is considered their own brother.

#### b. Antisymmetric:
A relation is **antisymmetric** if \( (a, b) \in R \) and \( (b, a) \in R \) implies \( a = b \). This means if \( a \) is the brother of \( b \), and \( b \) is the brother of \( a \), then \( a \) and \( b \) must be the same person.

- Antisymmetry does not hold, because two distinct people can be brothers. Therefore, **antisymmetry fails**.

#### c. Transitive:
As mentioned earlier, transitivity also does not hold.

Since \( R \) is neither reflexive, antisymmetric, nor transitive, **\( R \) is not a partial ordering**.

### Conclusion:
- **\( R \) is not an equivalence relation**.
- **\( R \) is not a partial ordering**.

Would you like to go into more details about any part of this analysis?

Here are 5 related questions to deepen understanding:
1. What is an example of a relation that is both symmetric and transitive but not reflexive?
2. Can you provide an example of a partial ordering relation in everyday life?
3. How would you define a strict partial ordering, and how does it differ from a regular partial ordering?
4. In what ways can we modify this relation to make it an equivalence relation?
5. How do equivalence relations partition a set, and can you give a real-world example?

**Tip:** Remember that partial orderings often arise in hierarchies or systems where some elements are comparable, but not all (e.g., family trees, job titles).

Let R be a relation of set A of people. R={(a, b) : a is brother of b}. Is it an equivalence relation? Is R partial ordering? (discard half brother, paternity brother)

Explore whether the brother relation R={(a, b) : a is brother of b} on a set of people is an equivalence relation or a partial ordering. Learn about the properties of reflexivity, symmetry, transitivity, and antisymmetry in set theory.

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