Definition 1: The collection S consists of pairwise disjoint nonempty subsets of
ID: 2965510 • Letter: D
Question
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 and every element of A belongs to exactly one subset in S.
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
Explanation / Answer
a) Given a collection S of subsets of A satisfying definition 1.
=> S consists of pairwise disjoint nonempty subsets of A..............................(i)
=> Every element of A belongs to a subset in S.....(ii)
(i) implies that S consists of nonempty subsets of A, hence satisfy the first condition of Defintion 2.
Also, no two subsets of A can have same element because they are disjoint. Hence, every element of A
belongs to exactly one susbet of A. Therefore, this makes S to satisfy second condition of Definition-2.
b) Given a collection S of subsets of A satisfying definition 2.
=> S consists of nonempty subsets of A..............................(i)
=> Every element of A belongs to exactly one subset in S.....(ii)
(i) automatically implies S consists of subsets of A, hence the first condition of Defintion 3.
(ii) condition is same for both the definition.
c) I think statement (c) is false, since S can contain an empty subset because definition 3 does not
impose the nonempty constraint to subsets. But Definition 1 requires subset to be nonempty.
Hence, if S satisfies Definition 3 it does not implies that it will satisfy Definition 1 because of the failure of
first condition of Definition 1.
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