Question

List all the partitions of the set \(A=\\{a, b, c\\}\). Compare your answer to the answer to Exercise 6 of Section 11.3 .

Short answer

The partitions of the set \(A = \{a, b, c\}\) are \(\{\{a\}, \{b\}, \{c\}\}\), \(\{\{a, b\}, \{c\}\}\), \(\{\{a, c\}, \{b\}\}\), \(\{\{b, c\}, \{a\}\}\), and \(\{\{a, b, c\}\}\)

Step-by-step solution

Understand the concept of partitions of a set

A partition of a set \(A\) is a set of nonempty subsets of \(A\) such that every element \(a\) in \(A\) is in exactly one of these subsets. In other words, each element of \(A\) is included in one and only one of these subsets.

List all the partitions of the set \(A\)

Now, we can list all the partitions of the set \(A = \{a, b, c\}\) as follows: \n1. \(\{\{a\}, \{b\}, \{c\}\}\) - Each element in its own subset. \n2. \(\{\{a, b\}, \{c\}\}\) - 'a' and 'b' in the same subset, 'c' in a different subset. \n3. \(\{\{a, c\}, \{b\}\}\) - 'a' and 'c' in the same subset, 'b' in a different subset. \n4. \(\{\{b, c\}, \{a\}\}\) - 'b' and 'c' in the same subset, 'a' in a different subset. \n5. \(\{\{a, b, c\}\}\) - All elements in the same subset.

Compare with the answer to Exercise 6 of Section 11.3

This step will require a copy of Exercise 6 of Section 11.3. With that, we can directly compare the listed partitions above to confirm they match the solution provided.