Question

Which of these collections of subsets are partitions of {−3,−2,−1, 0, 1, 2, 3}?

Answer 1

Answer:

a) Yes

b)No

c) Yes

d) No

Step-by-step explanation:

Remember, a collection $\mathcal{B}$ of subsets of a set $B$ is a partition of $B$ if the union of the subsets is $B$, that is, $\cup \mathcal{B}=B$ and the elements of $\mathcal{B}$ are disjoints.

Let $B=\{-3,-2,-1, 0, 1, 2, 3\}$

Then

a) $\{-3,-1, 1, 3\}\cup \{-2, 0, 2\} =B$ and $\{-3,-1, 1, 3\}\cap \{-2, 0, 2\}=\emptyset$.

Then the collection $\mathcal{B}=\{\{-3,-1, 1, 3\}, \{-2, 0, 2\}\}$ is a partition of B.

b) $\{-3,-2,-1, 0\}\cup \{0, 1, 2, 3\}=B$ and $\{-3,-2,-1, 0\}\cap \{0, 1, 2, 3\}=\{0\}$

Since the sets $\{-3,-2,-1, 0\},\{0, 1, 2, 3\}$ aren't disjoints then they aren't a partition of B.

c) $\{-3, 3\}\cup\{-2,2\}\cup\{-1,1\}\cup\{0\}=B$

and  

$\{-3, 3\}\cap\{-2, 2\}=\emptyset\\\{-3, 3\}\cap\{-1, 1\}=\emptyset\\\{-3, 3\}\cap\{0\}=\emptyset\\\{-2, 2\}\cap\{-1, 1\}=\emptyset\\\{-2, 2\}\cap\{0\}=\emptyset\\\{-1, 1\}\cap\{0\}=\emptyset$

Then the elements of the collection $\mathcal{B}=\{\{-3, 3\},\{-2, 2\},\{-1, 1\},\{0\}\}$ are disjoints.

Therefore, $\mathcal{B}$ is a partition of B.

d) $\{-3,-2, 2, 3\}\cup \{-1, 1\}\neq B$ because $0\in \{-3,-2, 2, 3\}\cup \{-1, 1\}$. Then the collection $\mathcal{B}=\{\{-3,-2, 2, 3\},\{-1, 1\}\}$ isn't a partition of B.