1. Understanding Set Notation
Set notation is a simple way of writing and describing sets. Sets are usually shown using capital letters like A, B or C. The elements of a set are written inside curly brackets { }. This helps express collections of numbers or objects quickly and clearly.
1.1. Basic Format
The most common way to write a set is:
\( A = \{1, 2, 3, 4\} \)
This means that the set A contains the elements 1, 2, 3 and 4.
1.2. Membership Symbols
- \( x \in A \) : x is an element of A
- \( y \notin A \) : y is not an element of A
2. Roster (Listing) Form
In this form, all elements are listed one by one inside braces. This works well when the set has only a few elements.
\( B = \{2, 4, 6, 8\} \)
2.1. Points to Note
- Elements are written without repetition.
- Order does not matter. \( \{1,2,3\} \) is the same as \( \{3,2,1\}. \)
- Each element must be clearly identifiable.
3. Set-Builder Form
Instead of listing elements, this form uses a condition or rule to define the set. It is helpful when dealing with large or infinite sets.
3.1. General Pattern
\( A = \{ x \mid x \text{ satisfies a certain property} \} \)
The symbol "\( \mid \)" is read as "such that".
3.2. Example
All even numbers greater than 10 can be written as:
\( E = \{ x \mid x \text{ is even and } x > 10 \} \)
4. Common Set Symbols
Some symbols are used very frequently when working with sets. These make mathematical expressions shorter and clearer.
4.1. Important Symbols
- \( \emptyset \) : empty set
- \( \subseteq \) : subset of
- \( \supseteq \) : superset of
- \( \subset \) : proper subset
- \( \cup \) : union
- \( \cap \) : intersection
- \( |A| \) : number of elements in set A (cardinality)
5. Interval Notation (Additional Useful Notation)
Interval notation is another way of writing sets, especially when dealing with ranges of numbers. It is commonly used for real numbers.
5.1. Basic Intervals
- \( (a, b) \) : all numbers between a and b, not including the endpoints
- \( [a, b] \) : all numbers between a and b, including both endpoints
- \( [a, b) \) : includes a but not b
- \( (a, b] \) : includes b but not a
5.2. Example
The set of real numbers greater than 3 and less than 8 can be written as:
\( (3, 8) \)