MHB Operations on Sets: Correcting Answers

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The discussion focuses on the properties of sets and their subsets, specifically regarding exhaustiveness and disjointness. Various subsets of a set A = {a, b, c} are evaluated for these properties, with some being identified as exhaustive and disjoint, while others are not. Additionally, conditions under which certain set operations yield specific results are analyzed, including intersections and unions involving universal and empty sets. There is some confusion regarding the specificity of the conditions requested, particularly concerning necessary and sufficient conditions. Overall, the conversation emphasizes understanding set operations and clarifying definitions related to subsets.
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just want to know if my answers are correct.

1. for any set A, a set of subsets of A is said to be exhaustive if the union of these subsets is A, and is said to be disjoint if no two of the subsets have any element in common. if $\displaystyle A\,=\,\{a,\,b,\,\,c\},\,$ tell whether the following set of subsets is exhaustive;disjoint.

a. $\{a\},\,\{b\}$ - disjoint
b. $\{a\},\,\{b,c\}$ - exhaustive and disjoint
c. $\{a,b\},\,\{b,c\}$ - exhaustive
d. $\{a\},\,\{a,b\}$ - neither
e. $\{a\},\,\{b\},\,\{c\}$ - exhaustive and disjoint

2. Tell under what conditions on the sets A and B we would have each of the following:

a. $\displaystyle A\cap B\,=\,\emptyset$ - if A & B are disjoint
b. $\displaystyle A\cap B\,=\,U$ - if both A & B are $\emptyset'$
c. $\displaystyle A\cup B\,=\,U$ - if A or B is $\emptyset'$
d. $\displaystyle A\cup B\,=\,\emptyset$ - if both A and B are $\emptyset$
e. $\displaystyle A\cap U\,=\,A$ - if $A\subset B$
f. $\displaystyle A\cup B\,=\,A$ -if $B\subset A$
g. $\displaystyle A\cap \emptyset\,=\,\emptyset$ - if A is $\emptyset$
h. $\displaystyle A\cap U\,=\,A$ - if A is $\emptyset$
i. $\displaystyle A\cup U\,=\,U$ - if $A\subset B$
j. $\displaystyle A\cup U\,=\,A$ - if A is $\emptyset$
k. $\displaystyle A\cup \emptyset\,=\,U$ - if A is $\emptyset'$
l. $\displaystyle A\cup\emptyset\,=\,\emptyset$ - if A is $\emptyset$

please tell me where I'm wrong and teach me how to approach that problem properly. thanks!:)
 
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bergausstein said:
1. for any set A, a set of subsets of A is said to be exhaustive if the union of these subsets is A, and is said to be disjoint if no two of the subsets have any element in common. if $\displaystyle A\,=\,\{a,\,b,\,\,c\},\,$ tell whether the following set subsets is exhaustive;disjoint.

a. $\{a\},\,\{b\}$ - disjoint
b. $\{a\},\,\{b,c\}$ - exhaustive and disjoint
c. $\{a,b\},\,\{b,c\}$ - exhaustive
d. $\{a\},\,\{a,b\}$ - neither
e. $\{a\},\,\{b\},\,\{c\}$ - exhaustive and disjoint
I agree.

bergausstein said:
2. Tell under what conditions on the sets A and B we would have each of the following
I find this question not specific enough. Do they want any of the possibly non-equivalent sufficient conditions, or any of the equivalent necessary and sufficient conditions? In any case, the latter answer is probably better.

I assume that $U$ denotes the universal set and $A'$ denotes the complement of $A$.

bergausstein said:
a. $\displaystyle A\cap B\,=\,\emptyset$ - if A & B are disjoint
Agree.
bergausstein said:
b. $\displaystyle A\cap B\,=\,U$ - if both A & B are $\emptyset'$
Agree. Note that $\emptyset'=U$.
bergausstein said:
c. $\displaystyle A\cup B\,=\,U$ - if A or B is $\emptyset'$
This is sufficient, but not necessary. $A\cup B=U$ happens iff, e.g., $B'\subseteq A$.
bergausstein said:
d. $\displaystyle A\cup B\,=\,\emptyset$ - if both A and B are $\emptyset$
Agree.
bergausstein said:
e. $\displaystyle A\cap U\,=\,A$ - if A\subset B
This answer does not make much sense because $B$ does not occur in the question. $A\cap U=A$ for all $A$.
bergausstein said:
f. $\displaystyle A\cup B\,=\,A$ -if B\subset A
Agree, but the subset can be improper (i.e., $B$ can equal $A$). In LaTeX it is usually denoted by \subseteq.
bergausstein said:
g. $\displaystyle A\cap \emptyset\,=\,\emptyset$ - if A is $\emptyset$
This happens for all $A$.
bergausstein said:
h. $\displaystyle A\cap U\,=\,A$ - if A is $\emptyset$
Same question as in e.
bergausstein said:
i. $\displaystyle A\cup U\,=\,U$ - if $A\subset B$
Same remark and answer as in e.
bergausstein said:
j. $\displaystyle A\cup U\,=\,A$ - if A is $\emptyset$
So, you think that the union of the empty set and everything is empty?
bergausstein said:
k. $\displaystyle A\cup \emptyset\,=\,U$ - if A is $\emptyset'$
Same remark as in b.
bergausstein said:
l. $\displaystyle A\cup\emptyset\,=\,\emptyset$ - if A is $\emptyset$
Agree.
 
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