Which of these collections of subsets are partitions of $\{1, 2, 3, 4, 5, 6\}$? For each collection that is not a partition, describe which properties of partitions do not hold. For each collection that is a partition, list the ordered pairs in the equivalence relation associated with the relation, and draw a graph representing that relation.

(a) $\{1, 2, 3\}, \{3, 4\}, \{4, 5, 6\}$

(b) $\{1, 4\}, \{2, 3, 6\}, \{5\}$

(c) $\{1, 5, 6\}, \{2, 3, 4\}$

(d) $\{1, 3, 5\}, \{2, 6\}$

<aside> 💡 (a) is not a partition of ****$\{1, 2, 3, 4, 5, 6\}$ since “$3$” in $\{1, 2, 3\}$ and $\{3, 4\}$

(b) is a partition of $\{1, 2, 3, 4, 5, 6\}$ and equivalence relation are $(1,4)$ for the first set, $(2,3),(2,6),(3,6)$ for the second subset:

(c) is a partition of $\{1, 2, 3, 4, 5, 6\}$ and equivalence relation are $(1,1),(1,5),(1,6),(5,1),(5,5),(5,6),(6,1),(6,5),(6,6)$ for the first set, $(2,2),(2,3),(2,4),(3,2),(3,3),(3,4),(4,2),(4,3),(4,4)$ for the second subset:

(d) is not a partition of $\{1, 2, 3, 4, 5, 6\}$ since “$4$” is missed.

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