Number of Elements in a Set: n(A)
The notation n(A) represents the number of elements contained in the set A. This is also called the cardinality of the set. For example, if A = {2, 4, 6, 8}, then n(A) = 4 because there are four elements in set A.
To find n(A), list all distinct elements in the set and count them carefully. If the set is described by a property, such as A = {x: x is a natural number less than 5}, then A = {1, 2, 3, 4} and n(A) = 4.
Example: Let B = {a, b, c, d, e}. Then n(B) = 5 because there are five elements in B.
- n(A) means the number of elements in set A.
- Count each distinct element once.
- If the set is defined by a rule, list elements to count them.
Complement of a Set: A'
The complement of a set A, written as A', consists of all elements in the universal set that are not in A. The universal set, often denoted by U, contains all possible elements under consideration for a particular problem.
For example, if the universal set U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}, then A' = {1, 3, 5} because these are the elements in U not in A.
To find A', first identify the universal set, then remove all elements of A from it.
- A' is the complement of A, elements not in A but in the universal set.
- Always define or know the universal set before finding complements.
- Example: If U = {1,2,3,4,5} and A = {2,3}, then A' = {1,4,5}.
Universal Set (U)
The universal set U contains all elements relevant to the discussion or problem. It is the 'whole' set from which subsets are taken.
When working with complements or Venn diagrams, the universal set forms the rectangle enclosing all other sets.
Example: If we are discussing natural numbers less than 10, then U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
- U is the universal set containing all elements under consideration.
- It is the reference set for complements and Venn diagrams.
- Always specify U clearly in problems involving complements.
Union of Sets: A ∪ B
The union of two sets A and B, written A ∪ B, is the set of all elements that are in A, or in B, or in both.
To find A ∪ B, combine all elements from both sets without repeating any element.
Example: If A = {1, 3, 5} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4, 5}. Notice that 3 appears in both sets but is listed only once in the union.
- A ∪ B includes every element in A or B or both.
- Do not repeat elements when listing the union.
- Use Venn diagrams to shade the union area (both circles combined).
Intersection of Sets: A ∩ B and Venn Diagrams
The intersection of two sets A and B, written A ∩ B, is the set of elements common to both A and B.
To find A ∩ B, list only those elements that appear in both sets.
Example: If A = {a, b, c} and B = {b, c, d}, then A ∩ B = {b, c}. These are the elements shared by both sets A and B.
- A ∩ B contains elements in both A and B.
- If no elements are common, A ∩ B is the empty set, denoted ∅.
- Venn diagrams show intersection as the overlapping region of two circles.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify sets | Write down elements or define by property | A = {1,2,3}, B = {2,3,4} | Clear set definition avoids confusion |
| Find n(A) | Count distinct elements in A | n(A) = 3 for A = {1,2,3} | Accurate counting earns marks |
| Find complement A' | List elements in U not in A | If U={1,2,3,4}, A={1,2}, then A'={3,4} | State universal set before complement |
| Find union A ∪ B | Combine all elements from A and B without repeats | A ∪ B = {1,2,3,4} for A={1,2}, B={2,3,4} | List all elements clearly |
| Find intersection A ∩ B | List elements common to both sets | A ∩ B = {2} for A={1,2}, B={2,3} | Show common elements only |
| Use Venn diagrams | Draw circles for sets inside rectangle for U | Shade union or intersection areas | Label sets and shade accurately for full marks |
Active recall
Quick check
1. If A = {1, 2, 3} and B = {3, 4, 5}, what is n(A)?
n(A) = 3 because there are three elements in set A.
2. Given U = {1, 2, 3, 4, 5} and A = {2, 4}, what is A'?
A' = {1, 3, 5}, the elements in U not in A.
3. Find A ∪ B if A = {a, b} and B = {b, c}.
A ∪ B = {a, b, c}, all elements from both sets without repeats.
4. Find A ∩ B if A = {x, y, z} and B = {w, x, y}.
A ∩ B = {x, y}, the elements common to both sets.