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).