Exercise 16 - Relation $R_2$ Analysis

We are given the set $M = \{1, 2, 3, 4, 5\}$, and the relation $R_2 \subset M \times M$ is defined as follows:

$R_2 = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 5)\}$

We need to analyze whether this relation is a partial order, a total order, or an equivalence relation. Let’s go step by step by analyzing the properties of the relation.

1. Reflexivity:

For a relation to be reflexive, each element of the set $M$ must be related to itself, i.e., $(a, a) \in R_2$ for all $a \in M$.

• The elements of $M$ are $\{1, 2, 3, 4, 5\}$.
• The only reflexive pair in $R_2$ is $(5, 5)$.
• However, the pairs $(1, 1), (2, 2), (3, 3), (4, 4)$ are not present in $R_2$.

Thus, $R_2$ is not reflexive.

2. Symmetry:

For a relation to be symmetric, if $(a, b) \in R_2$, then $(b, a)$ must also be in $R_2$ for all $a, b \in M$.

• In $R_2$, we have the pairs $(1, 2), (2, 3), (3, 4), (4, 5)$, but their symmetric counterparts $(2, 1), (3, 2), (4, 3), (5, 4)$ are not present.

Thus, $R_2$ is not symmetric.

3. Transitivity:

For a relation to be transitive, if $(a, b) \in R_2$ and $(b, c) \in R_2$, then $(a, c)$ must also be in $R_2$.

• From $(1, 2) \in R_2$ and $(2, 3) \in R_2$, we would expect $(1, 3)$ to be in $R_2$, but it is not.
• Similarly, from $(2, 3) \in R_2$ and $(3, 4) \in R_2$, we would expect $(2, 4)$ to be in $R_2$, but it is not.
• The same holds for other pairs.

Thus, $R_2$ is not transitive.

4. Antisymmetry:

For a relation to be antisymmetric, if $(a, b) \in R_2$ and $(b, a) \in R_2$, then it must hold that $a = b$.

• There are no pairs $(a, b)$ and $(b, a)$ for $a \neq b$ in $R_2$, so antisymmetry is vacuously satisfied.

Thus, $R_2$ is antisymmetric.

Conclusion:

• $R_2$ is not a partial order because it is not reflexive (even though it is antisymmetric).
• $R_2$ is not a total order because it is not reflexive and does not compare every element in $M$.
• $R_2$ is not an equivalence relation because it is neither reflexive, symmetric, nor transitive.

Summary of $R_2$ Properties:

• Not Reflexive.
• Not Symmetric.
• Not Transitive.
• Antisymmetric.

Would you like further explanation on this topic? Or perhaps assistance with the remaining parts of the exercise?

Here are 5 related questions for further exploration:

1. What conditions must a relation satisfy to be a partial order?
2. How can a relation be modified to become a total order?
3. Can you give an example of a reflexive, symmetric, and transitive relation?
4. Why is antisymmetry important in defining partial orders?
5. How do equivalence relations partition a set?

Tip: Antisymmetric relations ensure that if $a$ is related to $b$ and $b$ to $a$, they must be the same element. This is crucial in ordering relations to prevent circular dependencies.