Essential Rules and Useful Examples of Set Complements
In Maths, sets can be defined as a collection of well-defined objects or elements. A set can be represented by a capital letter symbol, and the number of elements in the finite set can be represented as the cardinal number of a set in a curly bracket {…}.
For example, set A is a collection of all the natural numbers, such as A equals {1, 2, 3, 4, 5, 6, 7, 8, ….. ∞}.
Sets can be represented in three forms:
Roster Form: Example - Set of even numbers less than 8 = {2, 4, 6}.
Statement Form: Example: A = {Set of Odd numbers less than 9}.
Set Builder Form: Example: A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}.
In this article, we are going to discuss the properties of the complement of a set, we are going to go through the properties of the complement of a set in brief.
What are the Types of Sets?
A set has many types, such as;
Empty Set or Null Set: It has no element present in it. Example: A = {} is a null set.
Finite Set: It has a limited number of elements. Example: A = {1, 2, 3, 4}.
Infinite Set: It has an infinite number of elements. Example: A = {x: x is the set of all whole numbers}.
Equal Set: Two sets which have the same members. Example: A = {1, 2, 5} and B = {2 , 5, 1}: Set A = Set B.
Subsets: A set ‘A’ is said to be a subset of B if each element of A is also an element of B. Example: A = {1, 2}, B = {1, 2, 3, 4}, then A ⊆ B.
Universal Set: A set that consists of all elements of other sets present in a Venn diagram. Example: A = {1, 2}, B = {2, 3}, The universal set here will be, U = {1, 2, 3}.
Properties of Complement of Set
There are three properties of the complement of a set. Let’s go through these three properties of the complement of a set:
Complement Laws: This is the first of the three properties of the complement of a set. The union of a set A and its complement denoted by A’ gives the universal set U of which A and A’ are a subset.
A ∪ A’ equals U
Also, the intersection of a set A and its complement A’ gives the empty set denoted by ∅.
A ∩ A’ = ∅
For Example: If U = {1 , 2 , 3 , 4 , 5 } and A = {1 , 2 , 3} then A’ = {4 , 5}. From this, it can be seen that A ∪ A’ = U = { 1 , 2 , 3 , 4 , 5}
Also, A ∩ A’ = ∅
Law of Double Complementation:
This is the second of the three properties of the complement of a set. According to this law of Double Complementation, if we take the complement of the complemented set named A’ then, we get the set A itself.
(A’ )’ equals A
In the previous example we can see that, if U = {1, 2, 3, 4, 5} and A = {1, 2, 3} then A’ = {4 , 5} . Now if take the complement of set A’ we get the following,
(A’ )’ = {1, 2, 3} = A , This gives us the set A itself.
Law of Empty Set and Universal Set:
According to this law, the complement of the universal set gives us the empty set and vice-versa that is,
∅’ equals U And U equals ∅’
These three are the properties of the complement of a set. These properties of the complement of a set are useful in Mathematics.
Solved Examples
Question 1) A universal set named U which consists of all the natural numbers which are multiples of the number 3, less than or equal to the number 20. Let set A be a subset of U which consists of all the even numbers and set B is also a subset of U consisting of all the prime numbers. Verify De Morgan Law.
Solution) We have to verify (A ∪ B)’ equals A’ ∩ B’ and (A ∩ B)’ equals A’∪B’. Given that, Using the properties of the complement of a set, let’s solve.
U equals {3, 6, 9, 12, 15, 18}
A equals {6, 12, 18}
B equals {3}
The union of both A and B can be given as,
A ∪ B equals {3, 6, 12, 18}
The complement of this union is given by,
(A ∪ B)’ equals {9, 15}
Also, the intersection and its complement are given by:
A ∩ B = ∅
(A ∩ B)’ equals {3, 6, 9, 12, 15,18}
Now, the complement of the set A and set B can be given as:
A’ = {3, 9, 15}
B’ = {6, 9, 12, 15, 18}
Taking the union of both these sets, we get,
A’∪B’ = {3, 6, 9, 12, 15, 18}
And the intersection of the complemented sets can be given as,
A’ ∩ B’ = {9, 15}
We can see that:
(A ∪ B)’ = A’ ∩ B’ = {9, 15}
And also,
(A ∩ B)’ = A’ ∪ B’ = {3, 6, 9, 12, 15,18}
Hence, the above-given result is true in general and is known as the De Morgan Law.
FAQs on Properties of Complement of a Set Explained
1. What are the main properties associated with the complement of a set?
The complement of a set has several fundamental properties that are crucial in set theory. The three primary properties are:
- Complement Laws: The union of a set A and its complement A' results in the universal set (A ∪ A' = U). Their intersection results in the empty set (A ∩ A' = ∅).
- Law of Double Complementation: The complement of a complemented set is the original set itself. This is represented as (A')' = A.
- Laws of Empty Set and Universal Set: The complement of the universal set (U) is the empty set (∅), and the complement of the empty set is the universal set. This is shown as U' = ∅ and ∅' = U.
2. How is the complement of a set represented symbolically and in a Venn diagram?
Symbolically, the complement of a set 'A' is most commonly denoted by A' (A prime) or sometimes as Aᶜ. In a Venn diagram, the complement of set A is visually represented by shading the entire region inside the universal set (U) that lies outside the circle representing set A. This shaded area contains all the elements that belong to U but do not belong to A.
3. What is the Law of Double Complementation and can you give an example?
The Law of Double Complementation states that if you take the complement of the complement of a set, you get the original set back. Symbolically, this is written as (A')' = A. For example, if the Universal Set U = {1, 2, 3, 4, 5} and set A = {1, 3}, then the complement of A is A' = {2, 4, 5}. If we then find the complement of A', which is (A')', we look for all elements in U that are not in A'. This gives us {1, 3}, which is the original set A.
4. Why is the concept of a Universal Set (U) essential when defining the complement of a set?
The Universal Set (U) is absolutely essential because the complement of a set A (A') is defined as the set of all elements that are not in A. Without the boundary of a Universal Set, the collection of 'not A' elements would be ambiguous and potentially infinite. The Universal Set provides the necessary context or 'universe' from which these elements are drawn. Therefore, A' specifically means 'all elements within U that are not in A'.
5. How do De Morgan's Laws connect the complement operation with set union and intersection?
De Morgan's Laws are a pair of transformative rules that create a powerful link between complements, unions, and intersections. They are essential for simplifying complex set expressions. The laws state:
- (A ∪ B)' = A' ∩ B': The complement of the union of two sets is equal to the intersection of their individual complements.
- (A ∩ B)' = A' ∪ B': The complement of the intersection of two sets is equal to the union of their individual complements.
These laws show how to distribute the complement operation 'over' union and intersection, effectively changing the primary operation from union to intersection, and vice versa.
6. How can you explain the property that the complement of the Universal Set is the Empty Set (U' = ∅)?
This property can be explained by definition. The Universal Set (U) contains all possible elements under consideration. The complement of U, denoted as U', is the set of elements that are in the universal set but also not in the universal set. Since there are no elements that can satisfy this condition, the resulting set contains no elements at all. The set with no elements is, by definition, the Empty Set (∅).
7. Can you provide a simple example to illustrate the complement laws?
Certainly. Let's define a Universal Set U = {a, b, c, d, e} and a set A = {a, c}.
- First, we find the complement of A, which is A' = {b, d, e}.
- Now, let's test the Complement Laws:
- A ∪ A' = {a, c} ∪ {b, d, e} = {a, b, c, d, e}, which is equal to the Universal Set (U).
- A ∩ A' = {a, c} ∩ {b, d, e} = { }, which is the Empty Set (∅) as there are no common elements.
8. What are some practical applications of understanding the properties of set complements?
Understanding the properties of set complements is fundamental in various fields beyond pure mathematics. For example:
- In Probability Theory, the complement is used to find the probability of an event not occurring. P(Not A) = 1 - P(A).
- In Computer Science and Logic, complements are the basis for the logical 'NOT' operation, which is critical in designing digital circuits, programming logic, and formulating database queries.
- In Data Analysis, it can be used to filter a dataset to find all entries that do not match a specific set of criteria.
