Operations on Sets: Correcting Answers

[bergausstein]

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!:)

 

[Evgeny.Makarov]

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.

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 $A$.

a. $\displaystyle A\cap B\,=\,\emptyset$ - if A & B are disjoint

Agree.

b. $\displaystyle A\cap B\,=\,U$ - if both A & B are $\emptyset'$

Agree. Note that $\emptyset'=U$.

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., $B'\subseteq A$.

d. $\displaystyle A\cup B\,=\,\emptyset$ - if both A and B are $\emptyset$

Agree.

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 $A$.

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.

g. $\displaystyle A\cap \emptyset\,=\,\emptyset$ - if A is $\emptyset$

This happens for all $A$.

h. $\displaystyle A\cap U\,=\,A$ - if A is $\emptyset$

Same question as in e.

i. $\displaystyle A\cup U\,=\,U$ - if $A\subset B$

Same remark and answer as in e.

j. $\displaystyle A\cup U\,=\,A$ - if A is $\emptyset$

So, you think that the union of the empty set and everything is empty?

k. $\displaystyle A\cup \emptyset\,=\,U$ - if A is $\emptyset'$

Same remark as in b.

l. $\displaystyle A\cup\emptyset\,=\,\emptyset$ - if A is $\emptyset$

Agree.