Given the following partition of the set A = { 6, 7, 8, 9, 10, 11 }. Find the associated relation { [6, 7, 8], [10], [9, 11] }.

What I understand is that a relation must be transitive, reflexive, and symmetric. How could I show that set A exhibits these characteristics?

To show that a set exhibits the characteristics of being transitive, reflexive, and symmetric, we need to examine the given partition and the associated relation.

1. Transitivity:
A relation is transitive if, for any three elements a, b, and c, such that (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.

In this case, we have three sets in the partition: [6, 7, 8], [10], and [9, 11]. Let's check if the relation is transitive by examining all possible pairs.

For the set [6, 7, 8]:
- (6, 7) is in the relation.
- (6, 8) is not in the relation.
- (7, 8) is not in the relation.

For the set [10]:
- There is only one element, so no pairs to consider.

For the set [9, 11]:
- (9, 11) is in the relation.

Since we have no pairs in the relation such that both (a, b) and (b, c) are in the relation, we don't have any violations of transitivity. Therefore, the relation associated with the given partition is transitive.

2. Reflexivity:
A relation is reflexive if every element in the set is related to itself.

In this case, we can see that each element in the set A = {6, 7, 8, 9, 10, 11} appears in at least one of the subsets in the partition. Therefore, every element is related to itself, and the relation associated with the partition is reflexive.

3. Symmetry:
A relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) is also in the relation.

In this case, let's consider the pairs in each subset:
For the set [6, 7, 8]:
- There are no pairs.

For the set [10]:
- There is only one element, so no pairs to consider.

For the set [9, 11]:
- There is only one pair: (9, 11).

Since we have no pairs other than (9, 11) in the relation, there are no pairs that violate symmetry. Hence, the relation associated with the partition is symmetric.

To summarize, the given condition { [6, 7, 8], [10], [9, 11] } satisfies the characteristics of being transitive, reflexive, and symmetric, and therefore, it represents a valid relation.