Given the following partition of the set A = { 6, 7, 8, 9, 10, 11 }. Find the associated relation { [6, 7, 8], [10], [9, 11] }.
To find the associated relation for the given partition of set A = { 6, 7, 8, 9, 10, 11 }, we need to understand what a partition and an associated relation are.
A partition of a set is a division of the set into non-empty and disjoint subsets, such that every element in the original set belongs to exactly one subset.
In this case, the given partition of the set A is { [6, 7, 8], [10], [9, 11] }. This means that the set A is divided into three subsets: [6, 7, 8], [10], and [9, 11]. These subsets are non-empty, which means they contain at least one element, and they are disjoint since they do not share any elements.
Now, let's find the associated relation for this partition. An associated relation is a relation that relates the elements of each subset in a partition. It tells us which elements are related to each other within the subsets.
Taking each subset at a time, we can create pairs of related elements. In this case, we have:
For the subset [6, 7, 8]:
- (6, 6), (6, 7), (6, 8), (7, 6), (7, 7), (7, 8), (8, 6), (8, 7), (8, 8)
For the subset [10]:
- (10, 10)
For the subset [9, 11]:
- (9, 9), (9, 11), (11, 9), (11, 11)
Combining all the pairs, we get the associated relation for the given partition: { (6, 6), (6, 7), (6, 8), (7, 6), (7, 7), (7, 8), (8, 6), (8, 7), (8, 8), (10, 10), (9, 9), (9, 11), (11, 9), (11, 11) }.
So, the associated relation for the given partition { [6, 7, 8], [10], [9, 11] } is { (6, 6), (6, 7), (6, 8), (7, 6), (7, 7), (7, 8), (8, 6), (8, 7), (8, 8), (10, 10), (9, 9), (9, 11), (11, 9), (11, 11) }.