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Mathematical procedures, unlike real-world operations, do not necessitate a distinct no-contamination room, surgical gloves, or masks. However, to find a solution, you’ll need problem-solving expertise, specific tools, strategies, and tricks, as well as a thorough understanding of all the fundamental principles. This is when the significance of Operations on Sets comes into play!
The concept of set operations is akin to fundamental operations on numbers. There are occasions when building a relationship between two or more sets is necessary. Then there’s the idea of set operations. Set union, set intersection, set complement, and set difference are the four main set operations. The numerous study material notes on Operations on Sets, notations for describing sets, how to work on sets, and their application in real life will be covered in this topic.
What are Set Operations?
A group of objects is referred to as a set. Each object in a set is referred to as an ‘Element.’ There are three ways to represent a set. Statement form, roster form, and set builder form are the three. Set operations are actions performed on two or more sets to create a relationship between them. The operations on Sets are divided into four categories:
- Union of sets
- Intersection of sets
- Complement of a set
- Difference between sets/Relative Complement
Basic Set Operations
Let’s go over each operation on Sets one by one now that we’ve covered the basics of a set and a Venn diagram. The following are the various set operations:
Union of Sets
A∪B (read as A union B) is the set of distinct items that belong to sets A and B or both for two given sets A and B. n(A∪B) = n(A) + n(B) − n(A∩B) where n(X) is the number of elements in set X, gives the number of elements in A ∪ B.
Consider the following example to understand better the set operation of the union of sets: If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then A ∪ B = {1, 2, 3, 4, 5, 6, 7} is the union of A and B.
Intersection of Sets
A∩B (read as A intersection B) is the set of common items that belong to both sets A and B for two given sets A and B. n(A∩B) = n(A)+n(B)−n(A∪B), where n(X) is the number of elements in set X, gives the number of elements in A∩B. Consider the following example to better understand the set operation of set intersection: The intersection of A and B is given by A ∩ B = {3, 4} if A = {1, 2, 3, 4} and B = {3, 4, 5, 7}.
Set Difference
Difference between sets is an operation on Sets that involves subtracting members from a set, which is related to the difference between numbers. All the elements in set A but not in set B are listed in the difference between sets A and B, marked as A − B. Consider the following example to better understand the set operation of set difference: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the final change between sets A and B is given by A – B = {1, 2}.
Complement of Sets
The complement of a set A indicated as A′ or Ac (read as A complement) is the set of all elements in the specified universal set(U) that are not present in set A. Consider the following example to better understand the complement of sets set operation: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4} then A’ = 5, 6, 7, 8, 9 is the complement of set A.
Important Notes on Set Operations
- For those who don’t know, the set operation formula for union of sets is n(A∪B) = n(A) + n(B) − n(A∩B) and on the other hand, the formula for the set operation for intersection of sets is n(A∩B) = n(A)+n(B)−n(A∪B).
- Any set can be combined with the universal set to form the universal set, and any set A can be intersected with the universal set to form set A.
- The operations on sets include union, intersection, difference, and complement.
- An empty set U′ = ϕ is the complement of a universal set. A universal set ϕ ′ = U is the complement of an empty set.
Conclusion
Operations on Sets is significant because it combines integer theory, axiom system models, infinite ordinals, and real numbers into a single coherent structure. Sets are used to hold a group of objects that are connected. They are significant in every branch of mathematics because sets are used or referred to in some way in every field of mathematics. They are necessary for the construction of more complicated mathematical structures.
