Set Operations
Set Operations can be defined as the operations performed on two or more sets to obtain a single set containing a combination of elements from all the sets being operated upon.
- Set operations are mathematical operations operated on sets, which are collections of distinct objects or elements.
- There are three major types of operation on sets, Union (∪), Intersection (∩), and Difference (-). Other operations include Difference, Complement, Addition & Subtraction
Set Operations
Union of Sets
Union of sets A and B, denoted by A ∪ B, is the set of distinct elements that belong to set A or set B, or both.
This operation can be represented as; A ∪ B = {x: x ∈ A or x ∈ B}
Where x is the elements present in both sets A and B.
Venn Diagram of A U B.
Venn diagram of A ∪ B
Above is the Venn Diagram of A U B.
Example:
Find the union of A = {2, 3, 4} and B = {3, 4, 5};
Solution:
A ∪ B = {2, 3, 4, 5}.
Intersection
The intersection of the sets A and B, denoted by A ∩ B, is the set of elements that belong to both A and B i.e. set of the common elements in A and B.
This operation is represented as: A∩B={x: x∈ A and x∈ B}
Here, x represents the elements that are common to both sets A and B.
Venn Diagram of A ∩ B
Venn diagram of A ∩ B
Example:
Find the intersection of A = {2, 3, 4} and B = {3, 4, 5}
Solution:
A ∩ B = {3, 4}.
The intersection of sets A and B, can also be interpreted as:
A ∩B = n (A) + n (B) – n (A∪B)
Where,
n(A)= the cardinality (or number of elements) of set A,
n(B)= the cardinality of set B,
n(A∪B) = the cardinality of the union of sets A and B.
Disjoint Set
Two sets are said to be disjoint if their intersection is the empty set. i.e, sets have no common elements.
Venn Diagram of A disjoint B
For Example: Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}
A and B are disjoint sets since both of them have no common elements.
Set Difference
The difference between sets is denoted by ‘A – B’, which is the set containing elements that are in A but not in B. i.e., all elements of A except the element of B.
Venn Diagram of A – B
Example: If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}, find A – B
Solution:
A – B = {1, 3, 5}
We can also state that the difference between set A and set B is equivalent to the intersection of set A with the complement of set B. Therefore,
A−B = A ∩ B′
where B′ = the complement of set B.
Complement of a Set
If U is a universal set and X is any subset of U, then the complement of X consists of all the elements in U that are not in X.
[Tex]X’ = \{a : a \in U \text{ and } a \notin X\}[/Tex]
Example:
Let U={1,2,3,4,5,6,7,8}
And A={1,2,5,6}
Then, the complement of A, denoted as A′, will be: A′={3,4,7,8}
Addition and Subtraction
The addition of sets A and B, known as Minkowski addition, results in a new set where each element is the sum of every possible pair of elements, one from set A and one from set B. In other words, the new set contains all possible sums of elements taken from the two sets. Set subtraction operates similarly but involves subtracting each element of set B from each element of set A. These operations are only meaningful when applied to numeric data types; if applied otherwise, they become symbolic with no practical significance.
Additionally, set addition is commutative, meaning the order of the sets does not affect the result. However, set subtraction is not commutative, as the order of the sets does affect the outcome.
n (A ∪ B) = n(A) + n(B) – n (A ∩ B)
A – B = A ∩ B’
Properties of Set Operations
Set operations have several important properties that govern their behavior. Here are some fundamental properties of set operations:
Closure Property
- Set operations are closed under their respective operations, meaning that performing an operation on sets results in another set.
- For example, the union, intersection, and difference of sets always produce sets as their results.
Commutative Property
- Union: A ∪ B = B ∪ A
- Intersection: A ∩ B = B ∩ A
- Symmetric Difference: A Δ B = B Δ A
Associative Property
- Union: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive Property
- Union over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Intersection over Union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Identity Property
- Union: A ∪ ∅ = A
- Intersection: A ∩ U = A, where U represents the universal set
- Symmetric Difference: A Δ ∅ = A
Complement Property
- Union: A ∪ A’ = U, where U is the universal set
- Intersection: A ∩ A’ = ∅ (the empty set)
Absorption Property
- Union over Intersection: A ∪ (A ∩ B) = A
- Intersection over Union: A ∩ (A ∪ B) = A
Examples of Operations on Sets
Example 1: Find the union of two sets A = {8, 10, 14} and B = {7, 16}
Solution:
(A ∪ B) = {8, 10, 14} ∪ {7, 16}
(A ∪ B) = {7, 8, 10, 14, 16}
Example 2: Find the intersection of sets P = {a, n, x} and Q = {x, y, z}
Solution:
(P ∩ Q) = {a, n, x} ∩ {x, y, z}
(P ∩ Q) = {x}
Example 3: Find the complement of set X = {4, 6, 9} where Universal set U = {1, 2, 3, 4, 6, 9}
Solution:
X’ = U – X
X’ = {1, 2, 3, 4, 6, 9} – {4, 6, 9}
X’ = {1, 2, 3}
Example 4: Given two sets A = {5, 6, 9, 10} and B = {3, 6, 12} then find, A – B and B – A
Solution:
A – B = {5, 6, 9, 10} – {3, 6, 12}
A – B = {5, 9, 10}
B – A = {3, 6, 12} – {5, 6, 9, 10}
B – A = {3, 12}
Example 5: Find the number of elements in set (A ∪ B) given that n(A) = 10, n(B) = 4 and n (A ∩ B) = 5.
Solution:
To find n (A ∪ B) we use formula:
n (A ∪ B) = n(A) + n(B) – n (A ∩ B)
n (A ∪ B) = 10 + 4 – 5
n (A ∪ B) = 9
Practice Questions on Set Operations
Q1. Find the union of two sets A = {2, 4, 5, 9} and B = {2, 6, 10, 12}
Q2. Find the intersection of sets P = {5, 6, 7, 9} and Q = {6, 9}
Q3. Find the complement of set X = {p, q, r} where Universal set U = {e, f, h, p, q, r}
Q4. Given two sets A = {1, 2, 4, 7, 10} and B = {1, 2, 8, 15} then find, B – A
Q5. Find the number of elements in set A given that n (A ∪ B) = 12, n(B) = 7 and n (A ∩ B) = 4.
Related Articles:
Can set operations be extended to more than two sets?
Yes, set operations can be extended to more than two sets. For example:
- The union of multiple sets: A ∪B ∪C
- The intersection of multiple sets: A ∩B ∩C
- The difference and complement operations can also be applied to multiple sets.
Can set operations be applied to non-numeric data?
Yes, set operations can be applied to any type of data, including numbers, letters, or even objects. The operations focus on the presence or absence of elements within sets, not the specific type of data.
What is a Venn diagram?
A Venn diagram is a visual representation of sets and their relationships. It uses circles to represent sets and shows their union, intersection, and difference through overlapping and non-overlapping areas.
How do set operations relate to real-world problems?
Set operations are widely used in data analysis, database queries, probability theory, and logic design. For example, the union of sets can help combine search results, while the intersection can find common factors in datasets.
How do set operations relate to logic gates?
This question deals with the application of set operations to digital circuits and logic gates, which is more of an advanced topic, specifically linking set theory with computer science or electrical engineering.
What is Power of a Set?
Set, including the empty set and the set itself, is called the power set of that set. the power set of a set with n elements will have 2 n subsets if the set has n elements.
