Question

Let \( D = \{a, b, c\} \). How many distinct ways can \( D \) be partitioned into non-empty subsets, representing equivalence relations?

• A. 2
• B. 3
• C. 5
• D. 6

Hint:

For finding distinct partitions of a set, you can use the Bell number or directly list the partitions for small sets.

Answer 1

The Correct Option is B

Let \( D = \{a, b, c\} \). How many distinct ways can \( D \) be partitioned into non-empty subsets, representing equivalence relations?

The number of distinct partitions of a set \( D \) into non-empty subsets is equivalent to the number of equivalence relations on \( D \). Each partition of the set corresponds to an equivalence relation because:

• Each equivalence relation on a set divides the set into equivalence classes (which are the subsets in a partition).
• Each partition of a set corresponds to a distinct equivalence relation.

We can solve this by determining the number of possible partitions of the set \( D = \{a, b, c\} \). The total number of partitions of a set is given by the Bell number for the set's size. For a set of size 3, the Bell number is 5. Hence, the set \( D = \{a, b, c\} \) has 5 distinct partitions. We can list and classify them as follows:

Step-by-Step Classification of Partitions:

• Partition 1: \( \{\{a\}, \{b\}, \{c\}\} \)

  This partition divides the set \( D \) into three subsets, each containing a single element. This corresponds to the equivalence relation where no elements are equivalent to each other.

• Partition 2: \( \{\{a, b\}, \{c\}\} \)

  This partition divides the set \( D \) into two subsets: one containing \( a \) and \( b \), and the other containing only \( c \). This corresponds to the equivalence relation where \( a \) and \( b \) are equivalent, but \( c \) is not equivalent to any other element.

• Partition 3: \( \{\{a, c\}, \{b\}\} \)

  This partition divides the set \( D \) into two subsets: one containing \( a \) and \( c \), and the other containing only \( b \). This corresponds to the equivalence relation where \( a \) and \( c \) are equivalent, but \( b \) is not equivalent to any other element.

• Partition 4: \( \{\{b, c\}, \{a\}\} \)

  This partition divides the set \( D \) into two subsets: one containing \( b \) and \( c \), and the other containing only \( a \). This corresponds to the equivalence relation where \( b \) and \( c \) are equivalent, but \( a \) is not equivalent to any other element.

• Partition 5: \( \{\{a, b, c\}\} \)

  This partition contains only one subset, which is the entire set \( D \). This corresponds to the equivalence relation where all elements of \( D \) are equivalent to each other.

Thus, the total number of distinct partitions of \( D \) is 5. These 5 partitions correspond to the 5 possible equivalence relations on the set \( D \).

Answer: There are 5 distinct ways to partition \( D = \{a, b, c\} \) into non-empty subsets, which is equivalent to the number of equivalence relations on \( D \).