Question

Let A = {a,b,c,d}. (a) List all partitions of A. (Hint: There are 15 total partitions of this set.) (b) How many of the partitions from part (a) have exactly 2 parts (or "blocks")? For example, {{a,c,d}, {b}}

has 2 parts.

Answer 1

For the set A = {a,b,c,d}, there are a total of 15 partitions. However, specifically for partitions with exactly 2 parts, there are 7 such partitions. Partitions divide the set into non-overlapping subsets, with each element being included in exactly one subset.

Step-by-step explanation:

When considering the set A = {a,b,c,d}, there are 15 total possible partitions of this set. A partition of a set is a way of dividing the set into non-empty, non-overlapping subsets, where each element of the original set is included in exactly one of the subsets. Since there are 15 and this is quite a large number, I will not list them all.

If we specifically look at partitions of set A that have exactly 2 parts, we can create partitions by pairing any one element with another, then place the leftover elements together. For example:

• {{a}, {b,c,d}}
• {{b}, {a,c,d}}
• {{c}, {a,b,d}}
• {{d}, {a,b,c}}
• {{a,b}, {c,d}}
• {{a,c}, {b,d}}
• {{a,d}, {b,c}}

These 7 partitions are the ones with exactly 2 parts. Therefore, there are 7 different partitions of set A that consist of 2 parts.