1. Meaning of a Set
A set is just a collection of objects or items that are grouped together because they share some common idea. The items inside a set are called elements. A set should always be clear — anyone should be able to say whether something belongs to it or not.
Sets are useful because they help us organise numbers, letters, or any objects in a simple and clean way.
1.1. Quick Examples
- Set of vowels: \( \{a, e, i, o, u\} \)
- Set of even numbers less than 10: \( \{2, 4, 6, 8\} \)
- Set of days starting with 'S': \( \{Saturday, Sunday\} \)
2. How We Write Sets
Sets are usually shown using capital letters like \(A, B, C\). Elements are written inside curly brackets { }. To show whether something is inside a set, we use:
- \( x \in A \): x is an element of A
- \( y \notin A \): y is not an element of A
2.1. Roster (Listing) Form
All elements are written one by one.
Example: \( A = \{1, 3, 5, 7\} \)
2.2. Set-Builder Form
We describe the property of the elements instead of listing them.
Example: \( A = \{ x \mid x \text{ is an odd number less than } 10 \} \)
3. Well-Defined Sets
A set is called well-defined if it is completely clear who belongs to it. There should be no confusion or opinion involved.
3.1. Examples
- Well-defined: Set of months with 30 days
- Not well-defined: Set of ‘nice songs’ (because different people have different opinions)
4. Some Basic Terms
These are small but important words used when working with sets.
4.1. Element
An element is just an item inside a set.
Example: In \( \{2,4,6\} \), 4 is an element.
4.2. Cardinality
Cardinality means the number of elements in a set. It is written as \(|A|\).
Example: If \( A = \{a,b,c,d\} \), then \(|A| = 4\).
4.3. Empty (Null) Set
This is a set with no elements. It is shown as \( \emptyset \) or \( \{\} \).
4.3.1. Examples
- Set of odd numbers divisible by 2
- Set of natural numbers less than 1
5. Quick Look at Types of Sets
Here are some very common types of sets that appear often. These will be studied in detail later, but this gives a quick idea.
5.1. Finite and Infinite
A finite set has a countable number of elements. An infinite set goes on forever.
Examples: \( \{1,2,3,4,5\} \) is finite, while \( \{1,2,3,\ldots\} \) is infinite.
5.2. Equal and Equivalent
Equal sets: Have the exact same elements.
Equivalent sets: Have the same number of elements (even if the actual items are different).
5.3. Singleton Set
A set with only one element.
Example: \( \{9\} \)
6. Representing Sets Visually
Sets are often shown using Venn diagrams. These help us see how sets overlap or relate to each other. You will use these when studying operations on sets.