Set and operation
A set is a collection of distinct elements or objects, grouped together. These elements can be anything - numbers, letters, words, or even other sets. The operation on sets refers to the different actions or manipulations that can be performed on sets.
There are several fundamental operations on sets, including:
1. Union: Given two sets A and B, the union operation (represented by the symbol ∪) combines all the elements from both sets into a new set. The resulting set will contain all the elements that are in A or B, or both.
Example: A = {1, 2, 3}, B = {3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5}
2. Intersection: Given two sets A and B, the intersection operation (represented by the symbol ∩) creates a new set that contains only the elements that are common to both sets.
Example: A = {1, 2, 3}, B = {3, 4, 5}
A ∩ B = {3}
3. Difference: Given two sets A and B, the difference operation (represented by the symbol \ or -) creates a new set that contains all the elements that are in A but not in B.
Example: A = {1, 2, 3}, B = {3, 4, 5}
A \ B = {1, 2}
4. Complement: In set theory, the complement of a set A (denoted by A') refers to all the elements that are not in A but exist in the universal set U. The universal set is typically the set that contains all the possible elements under consideration.
Example: A = {1, 2, 3}, U = {1, 2, 3, 4, 5}
A' = {4, 5}
These are the basic set operations, but there are more advanced operations like symmetric difference, Cartesian product, and power set, which involve combining or manipulating sets in different ways.