Question

How many distinct pairs of disjoint non-empty subsets of A are there, the union of which is all of A? A {0,4,5,7}

Answer 1

Let's consider the set A with elements {0, 4, 5, 7}.

We want to find out how many distinct pairs of disjoint (non-intersecting) non-empty subsets of A there are, such that the union of each pair of subsets is the entire set A.

Here's how we can approach the problem:

1. Each element of A must belong to one and only one of the two subsets in each distinct pair. This is because the subsets are disjoint (they share no common elements) and their union must be A.

2. Since A has 4 elements, each element has a choice: it can go into the first subset or the second subset. This creates a binary choice for each element.

3. Because there are 4 binary choices (one for each element), and 2 options for each choice, there are a total of 2^4 = 16 possible ways to divide the elements of A into two groups (including groupings where one group may be empty).

4. However, we cannot have a pair where one subset is empty since we're looking for non-empty subsets. There are 2 cases where we have an empty subset: one in which all elements are in the first subset and the other in which all elements are in the second subset. We have to subtract these 2 cases from the total, leaving us with 16 - 2 = 14 valid combinations.

5. Finally, because the order of the subsets in a pair doesn't matter (i.e., the pair (subset1, subset2) is the same as (subset2, subset1)), we've counted each valid pair twice. To correct for this, we divide the number of valid combinations by 2.

Therefore, the total number of distinct pairs of non-empty disjoint subsets of A whose union equals A is (16 - 2) / 2 = 14 / 2 = 7.

So, there are 7 distinct pairs of disjoint non-empty subsets of A, the union of which is all of A.

Answer 2

Let set A be written as B∪C, where B and C are 2 disjoint nonempty subsets of A.

the number of elements of B and C respectively can be:

i) 1, 3
ii)2, 2

for case i we have the following pairs of sets:

{0}, {4,5,7}

{4}, {0,5,7}

{5}, {0,4,5}

{7}, {0,4,5}

for case ii, consider only one of the sets, which can be any of these:

{0,4}, {0,5}, {0,7}, {4,5}, {4,7},{5,7}

Clearly these pairs complement each other, that is if the first set is {0,4}, the second set is {5,7}, if the first set is {0,5} the second set is {4,7}, and if the first set is {0,7} the second is {4,5}.

Thus there are 4+3=7 pairs of subsets


Answer: 7