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}

Answer:a) Yesb)Noc) Yesd) NoStep-by-step explanation:Remember, a collection [tex]\mathcal{B}[/tex] of subsets of a set [tex]B[/tex] is a partition of [tex]B[/tex] if the union of the subsets is [tex]B[/tex], that is, [tex]\cup \mathcal{B}=B[/tex] and the elements of [tex]\mathcal{B}[/tex] are disjoints.Let [tex]B=\{-3,-2,-1, 0, 1, 2, 3\}[/tex]Thena) [tex]\{-3,-1, 1, 3\}\cup \{-2, 0, 2\} =B[/tex] and [tex]\{-3,-1, 1, 3\}\cap \{-2, 0, 2\}=\emptyset[/tex].Then the collection [tex]\mathcal{B}=\{\{-3,-1, 1, 3\},  \{-2, 0, 2\}\}[/tex] is a partition of B.b) [tex]\{-3,-2,-1, 0\}\cup \{0, 1, 2, 3\}=B[/tex] and [tex]\{-3,-2,-1, 0\}\cap \{0, 1, 2, 3\}=\{0\}[/tex]Since the sets [tex]\{-3,-2,-1, 0\},\{0, 1, 2, 3\}[/tex] aren't disjoints then they aren't a partition of B.c) [tex]\{-3, 3\}\cup\{-2,2\}\cup\{-1,1\}\cup\{0\}=B[/tex]and  [tex]\{-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[/tex] Then the elements of the collection [tex]\mathcal{B}=\{\{-3, 3\},\{-2, 2\},\{-1, 1\},\{0\}\}[/tex] are disjoints.Therefore, [tex]\mathcal{B}[/tex] is a partition of B.d) [tex]\{-3,-2, 2, 3\}\cup \{-1, 1\}\neq B[/tex] because [tex]0\in \{-3,-2, 2, 3\}\cup \{-1, 1\}[/tex]. Then the collection [tex]\mathcal{B}=\{\{-3,-2, 2, 3\},\{-1, 1\}\}[/tex] isn't a partition of B.