1. What Is a Singleton Set?
A singleton set is a set that contains exactly one element. Nothing more than one, nothing less. It is the smallest possible non-empty set.
Because it has only one element, it is very simple to work with and shows up often in examples and definitions.
1.1. Notation
A singleton set is written just like any other set, but with only one item inside the curly brackets:
\( \{a\} \)
2. Examples of Singleton Sets
Here are some common examples where only one element is present:
2.1. Simple Examples
- \( \{5\} \)
- \{“blue”\}
- \{\pi\}
- Set of letters in the word “Sky” → \{S, k, y\} (no singleton here) but set of vowels in “Sky” → \{\} (empty)
2.2. Examples Using Conditions
\( A = \{ x \mid x \text{ is the only even prime number} \} = \{2\} \)
\( B = \{ x \mid x^2 = 49 \text{ and } x > 0 \} = \{7\} \)
3. Cardinality of a Singleton Set
Since a singleton set has exactly one element, its cardinality is 1.
\( |\{x\}| = 1 \)
4. Important Points
Some things to remember about singleton sets:
4.1. Key Notes
- A singleton set is not the same as an empty set.
- \( \{0\} \) is a singleton set, not empty.
- A singleton set can contain numbers, letters, symbols, or even another set as its only element.
- It is the simplest type of non-empty set.
5. Why Singleton Sets Matter
Singleton sets help describe functions, mappings, subsets, and many mathematical properties. They are also useful when checking if a condition leads to exactly one solution.