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

Given the set \( D = \{a, b, c\} \), determine the number of unique ways to divide \( D \) into non-empty subsets, where each division represents an equivalence relation.

The number of distinct ways to partition a set \( D \) into non-empty subsets is identical to the number of equivalence relations on \( D \). This equivalence exists because:

• Every equivalence relation on a set inherently divides that set into equivalence classes, which are the non-empty subsets of a partition.
• Conversely, any partition of a set uniquely defines an equivalence relation.

To solve this, we must find the total number of possible partitions for the set \( D = \{a, b, c\} \). This quantity is known as the Bell number corresponding to the set's cardinality. For a set with 3 elements, the Bell number is 5. Therefore, the set \( D = \{a, b, c\} \) can be partitioned in 5 distinct ways. These partitions are enumerated and categorized below:

Categorization of Partitions:

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

  This partition separates \( D \) into three singleton subsets, representing an equivalence relation where no elements are considered equivalent to each other.

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

  This partition divides \( D \) into two subsets: \( \{a, b\} \) and \( \{c\} \). It corresponds to an 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 \( D \) into two subsets: \( \{a, c\} \) and \( \{b\} \). It corresponds to an 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 \( D \) into two subsets: \( \{b, c\} \) and \( \{a\} \). It corresponds to an equivalence relation where \( b \) and \( c \) are equivalent, but \( a \) is not equivalent to any other element.

• Partition 5: \( \{\{a, b, c\}\} \)

  This partition consists of a single subset encompassing the entire set \( D \). It represents an equivalence relation where all elements of \( D \) are equivalent to each other.

Consequently, there are 5 distinct partitions for \( D \), each corresponding to a unique equivalence relation on \( D \).

Answer: The set \( D = \{a, b, c\} \) can be partitioned into non-empty subsets in 5 distinct ways, which is equivalent to the number of possible equivalence relations on \( D \).