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

Question

Which of these collections of subsets are partitions of {−3,−2,−1, 0, 1, 2, 3}?
a) {−3,−1, 1, 3}, {−2, 0, 2}
b) {−3,−2,−1, 0}, {0, 1, 2, 3}
c) {−3, 3}, {−2, 2}, {−1, 1}, {0}
d) {−3,−2, 2, 3}, {−1, 1}

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Adalynn 4 days 2021-10-11T19:14:06+00:00 1 Answer 0

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    2021-10-11T19:15:18+00:00

    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.

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