1. Meaning of an Empty Set
An empty set is a set that has no elements in it. There is nothing inside the set. It is also called the null set.
This is one of the simplest ideas in set theory, yet very important because it shows up in many problems where no object meets the given condition.
1.1. Notation
The empty set is written in two ways:
- \( \emptyset \)
- \( \{\} \)
Both mean the same: a set with zero elements.
2. Examples of Empty Sets
An empty set appears whenever no object satisfies the given condition. Here are some common examples:
2.1. Simple Examples
- Set of whole numbers less than 0
- Set of even numbers between 5 and 7
- Set of prime numbers between 8 and 10
- Set of letters in the word “SKY” that are vowels
2.2. Examples Using Math Notation
Sometimes the definition makes it clearer:
\( A = \{ x \mid x^2 = -4 \} \Rightarrow A = \emptyset \)
\( B = \{ x \mid x \text{ is an odd multiple of 2} \} \Rightarrow B = \emptyset \)
3. Cardinality of the Empty Set
The empty set has no elements, so its cardinality is zero.
\( |\emptyset| = 0 \)
4. Important Notes
Some important things to remember about the empty set:
4.1. Key Points
- The empty set is a subset of every set.
- There is only one empty set. Different empty-looking sets are actually the same.
- Having \(0\) as an element is not the same as being empty. Example: \( \{0\} \) is not empty; it has one element.
- Empty sets often appear when applying conditions that cannot be satisfied.
5. Why Empty Sets Matter
They help in many areas of mathematics, especially while checking for solutions, defining subsets, and working with set operations. Even though the empty set has no elements, it still behaves like a proper set and follows all set rules.