Question

1.58. It was mentioned that there are three ways that a collection S of subsets of a nonempty set A is defined to be a partition of A. Definition 1 The collection S consists of pairwise disjoint nonempty subsets of A and every element of A belongs to a subset in S. Definition 2 The collection S consists of nonempty subsets of A and every element of A belongs to exactly one subset in S. Definition 3 The collection S consists of subsets of A satisfying the three properties (1) every subset in S is nonempty, (2) every two subsets of A are equal or disjoint and (3) the union of all subsets in S is A. (a) Show that any collection S of subsets of A satisfying Definition 1 satisfies Definition 2. (b) Show that any collection S of subsets of A satisfying Definition 2 satisfies Definition 3. (c) Show that any collection S of subsets of A satisfying Definition 3 satisfies Definition 1.

          1.58. It was mentioned that there are three ways that a collection S of subsets of a nonempty set A is defined to be a partition of A.
Definition 1 The collection S consists of pairwise disjoint nonempty subsets of A and every element of A belongs to a subset in S.
Definition 2 The collection S consists of nonempty subsets of A and every element of A belongs to exactly one subset in S.
Definition 3 The collection S consists of subsets of A satisfying the three properties (1) every subset in S is nonempty, (2) every two subsets of A are equal or disjoint and (3) the union of all subsets in S is A.
(a) Show that any collection S of subsets of A satisfying Definition 1 satisfies Definition 2.
(b) Show that any collection S of subsets of A satisfying Definition 2 satisfies Definition 3.
(c) Show that any collection S of subsets of A satisfying Definition 3 satisfies Definition 1.

1.58. It was mentioned that there are three ways that a collection S of subsets of a nonempty set A is defined to be a partition of A.
Definition 1 The collection S consists of pairwise disjoint nonempty subsets of A and every element of A belongs to a subset in S.
Definition 2 The collection S consists of nonempty subsets of A and every element of A belongs to exactly one subset in S.
Definition 3 The collection S consists of subsets of A satisfying the three properties (1) every subset in S is nonempty, (2) every two subsets of A are equal or disjoint and (3) the union of all subsets in S is A.
(a) Show that any collection S of subsets of A satisfying Definition 1 satisfies Definition 2.
(b) Show that any collection S of subsets of A satisfying Definition 2 satisfies Definition 3.
(c) Show that any collection S of subsets of A satisfying Definition 3 satisfies Definition 1.

Added by Michelle M.

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