You read a complementary set, denoted as Ac or A', as "the complement of A." It represents all elements in the universal set (U) that are not in set A.
Understanding Complementary Sets
To fully grasp how to "read" a complementary set, let's break down the concept:
- Universal Set (U): This is the overarching set that contains all possible elements under consideration. Think of it as the "universe" within which you're working.
- Set A: This is a specific set within the universal set.
- Complement of A (Ac or A'): This is the set containing every element that is in the universal set (U) but not in set A.
Expressing the Complement Mathematically
The complement of A is mathematically expressed as:
Ac = {x ∈ U | x ∉ A}
This reads as: "A complement is the set of all x belonging to U such that x does not belong to A."
Examples
Here are some examples to illustrate how to read and understand complementary sets:
Example 1:
- U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (The universal set is the set of integers from 1 to 10)
- A = {2, 4, 6, 8, 10} (Set A is the set of even numbers from 1 to 10)
Then, Ac = {1, 3, 5, 7, 9} (The complement of A is the set of odd numbers from 1 to 10).
You would read Ac as "the complement of A," which in this case is the set containing elements that are in U but not in A (i.e., the odd numbers).
Example 2:
- U = {a, b, c, d, e}
- B = {b, d}
Then, B' = {a, c, e} (The complement of B is the set containing a, c, and e).
You would read B' as "the complement of B," which includes all elements from U that are not found within B.
Key Takeaways
- When you see Ac or A', say "the complement of A."
- The complement always refers to the elements outside the original set but within a defined universal set.
- Understanding the universal set is crucial for correctly identifying the complement.
In essence, reading "the complement of A" signifies that you're considering all elements within the universe that are not members of set A.
