Basic Set Terminology and Notation
A set is a collection of distinct objects, called elements. We use curly braces { } to list elements or describe them by a property.
The number of elements in a set A is written as n(A). For example, if A = {2, 4, 6}, then n(A) = 3.
To indicate that an element x belongs to a set A, we write x ∈ A. If x is not in A, we write x ∉ A. For instance, if A = {1, 3, 5}, then 3 ∈ A but 2 ∉ A.
- n(A): number of elements in set A
- x ∈ A: x is an element of A
- x ∉ A: x is not an element of A
Universal Set, Empty Set, and Complements
The universal set, denoted by U, contains all elements under consideration in a particular context. For example, if we are discussing natural numbers less than 10, then U = {1, 2, 3,..., 9}.
The empty set, denoted by ∅, contains no elements. It is a subset of every set.
The complement of a set A, written A', consists of all elements in U that are not in A. For example, if U = {1, 2, 3, 4, 5} and A = {2, 4}, then A' = {1, 3, 5}.
- U: universal set containing all relevant elements
- ∅: empty set with no elements
- A': complement of A, elements in U not in A
Subsets and Their Notation
A set A is a subset of set B, written A ⊆ B, if every element of A is also an element of B. For example, if A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
If A is not a subset of B, we write A ⊈ B. For instance, if A = {1, 4} and B = {1, 2, 3}, then A ⊈ B because 4 ∉ B.
- A ⊆ B: every element of A is in B
- A ⊈ B: A is not a subset of B
Union and Intersection of Sets
The union of two sets A and B, written A ∪ B, is the set of elements that are in A, or in B, or in both. For example, if A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}.
The intersection of two sets A and B, written A ∩ B, is the set of elements common to both A and B. Using the same sets, A ∩ B = {2}.
These operations can be extended to three sets, for example, A ∪ B ∪ C and A ∩ B ∩ C.
- A ∪ B: elements in A or B or both
- A ∩ B: elements in both A and B
Using Venn Diagrams to Illustrate Sets and Their Relationships
Venn diagrams use circles to represent sets within a rectangle representing the universal set U.
For two sets, two overlapping circles show the union (all areas covered by either circle) and the intersection (the overlapping area).
For three sets, three overlapping circles create regions representing all possible intersections and unions, including elements common to all three sets or unique to one set only. This visualisation helps understand complex relationships and solve problems involving sets.
- Circles represent sets inside universal set rectangle
- Overlap shows intersection
- All covered areas show union
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify sets and universal set | Define each set clearly and specify the universal set | A = {x: x is a natural number ≤ 5}, U = {1, 2, 3, 4, 5} | Ensures clarity in notation and correct use of U and sets |
| Determine membership | Check if elements belong to sets using ∈ or ∉ | 3 ∈ A, 6 ∉ A | Use correct symbols to state membership or non-membership |
| Find subsets | Check if all elements of A are in B to write A ⊆ B or A ⊈ B | A = {1, 2}, B = {1, 2, 3}, so A ⊆ B | Accurately state subset relationships for marks |
| Calculate union and intersection | Combine elements for union; find common elements for intersection | A ∪ B = {1, 2, 3}, A ∩ B = {2} | Show working clearly to justify answers |
| Draw Venn diagrams | Represent sets and their relationships visually | Two overlapping circles for A and B with shaded regions | Use diagrams to support explanations or solve problems |
Active recall
Quick check
1. If A = {1, 3, 5} and U = {1, 2, 3, 4, 5}, what is A'?
A' = {2, 4} because these are elements in U not in A.
2. Is A = {2, 4} a subset of B = {1, 2, 3, 4, 5}?
Yes, A ⊆ B since every element of A (2 and 4) is in B.
3. Find A ∪ B if A = {1, 2} and B = {2, 3}.
A ∪ B = {1, 2, 3} includes all elements from both sets without repetition.
4. What does the intersection A ∩ B represent?
A ∩ B is the set of elements common to both A and B.