Introduction to Sets
Xcel Learning Notes
Definition of a Set
A set is a well-defined collection of objects or members. These objects are referred to as elements. In mathematics, we use curly brackets { } to enclose the members of a set.
• A set must be well-defined so that we can clearly determine if an object belongs to it or not.
• Elements are usually separated by commas within the brackets.
Finite and Infinite Sets
Sets are categorized based on the number of elements they contain.
• Finite Sets: These are sets that contain a specific number of elements that can be counted or listed. For example, the set of days in a week is finite because it has exactly 7 members.
• Infinite Sets: These are sets that contain so many members that it is impossible to count them all. They go on forever. An example would be the set of all counting numbers {1, 2, 3, …}.
The Empty Set (Null Set)
An empty set is a unique set that contains no elements or members at all.
• It is also commonly referred to as a “null” set.
• Notation: It is represented by either a pair of empty curly braces { } or the symbol Ø.
Key Formula
A = { } OR A = Ø
⚠ Common Mistake
Do not combine the symbols. Writing {Ø} is incorrect because it represents a set containing the null symbol as an element, rather than an empty set.
Universal Sets
A universal set (U) is the set that contains all possible members under consideration for a particular problem or group.
• Any other set being discussed within that context will be a subset of the universal set.
• For example, if we are discussing ‘Apples’, ‘Oranges’, and ‘Bananas’, the Universal Set might be ‘All Fruits’.
Key Formula
U = {all members of a group}
📌 Exam Tip
In Venn Diagrams, the Universal Set is always represented by the rectangle surrounding the circles.
Disjoint Sets
Disjoint sets are sets that have absolutely no elements in common.
• If you compare the members of two disjoint sets, you will not find any matches.
• The intersection of disjoint sets results in an empty set.
• On a Venn Diagram, disjoint sets are drawn as circles that do not touch or overlap.
Key Formula
A ∩ B = Ø
📌 Exam Tip
If the question states that two sets are disjoint, ensure you draw them separately with no overlapping area.
✏️ Practice Questions
- State whether the set of ‘Prime Numbers’ is finite or infinite.
- Define the term ‘Empty Set’ and provide both symbols used to represent it.
- If Set A = {stars in the sky}, classify this set as finite or infinite.
- Set X = {2, 4, 6} and Set Y = {1, 3, 5}. Are these sets disjoint? Explain your answer.
- Suggest a possible Universal Set for the following subsets: {Jazz, Reggae, Pop}.
Xcel Learning Notes — xcellearning.com
