Finding all partitions of two sets

Question

When trying to find all the partitions of a set; there should always be at least one partition (the original set itself)?

I'm trying to confirm this by listing all the partitions of these two sets...

1. First set: A = {a, b, c}

P(A) = {{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

Partition: {{a}, {b}, {c}}

Partition: {{a}, {b, c}}

Partition: {{b}, {a, c}}

Partition: {{c}, {a, b}}

Partition: {{a, b, c}}

2. Second set: B = {1, 2}

P(B) = {{}, {1}, {2}, {1, 2}}

Partition: {{1}, {2}}

Partition: {{1, 2}}

Answer 1

Yes, this looks correct.

One quick way to check if you've listed all of them is to check if the number of partitions of your $n$-element set is equal to the $n^{th}$ Bell Number.