Question

If X is any set, define P₂(X) to be the set of all two-element subsets of X. For example, if X={a,b,c,d}, then P₂(X) is the set {{a,b},{a,c},{a,d},{b,c},{b,d},{c,d}}, containing six subsets. We will say that a subset Y of P₂(X) covers X if every element of X is in exactly one of the subsets in Y. For X={a,b,c,d}, the subset {{a,b},{c,d}} covers X. Show that P₂(X) can be divided into three disjoint subsets, each of which covers X.

Answer 1

To divide P₂(X) into three disjoint subsets that each cover X for set {a, b, c, d}, we form three non-overlapping groups of pairs, ensuring each element of X is included exactly once within each group.

Step-by-step explanation:

To show that P₂(X) can be divided into three disjoint subsets, each of which covers X, we start by considering a set X with an unspecified number of elements. Let's take X to be {a, b, c, d} as given in the example. If each two-element subset of X is formed, we have the following pairs:

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

We can arrange these pairs into three groups:

• Group 1: {{a, b}, {c, d}}
• Group 2: {{a, c}, {b, d}}
• Group 3: {{a, d}, {b, c}}

Each group forms a disjoint subset of P₂(X) and covers the set X because every element of X is included in exactly one pair within a group. Therefore, P₂(X) can be divided into three subsets that each cover X with no overlap between the subsets.