Operations on sets

Operations on Sets - Overview and Basic Definitions

Operations on sets are procedures that combine or modify sets to form new sets. They are fundamental in set theory and are widely used in mathematics, computer science, and logic.

Basic Set Operations

Here are the most common operations on sets with brief explanations:

• Union (∪): The union of two sets $A$ and $B$ is the set of all elements that are in $A$ or $B$ or in both.

$A\cup B=\{x:x\in A\text{ or }x\in B\}$

• Intersection (∩): The intersection of two sets $A$ and $B$ is the set of all elements that are in both $A$ and $B$.

$A\cap B=\{x:x\in A\text{ and }x\in B\}$

• Difference (−): The difference of two sets $A$ and $B$ (also called the relative complement) is the set of all elements that are in $A$ but not in $B$.

$A-B=\{x:x\in A\text{ and }x\notin B\}$

• Complement (′): The complement of a set $A$ relative to a universal set $U$ is the set of all elements in $U$ that are not in $A$.

$A^{\prime }=U-A=\{x:x\in U\text{ and }x\notin A\}$

Links to Detailed Operations

• Union of Sets
• Intersection of Sets
• Difference of Sets
• Complement of a Set

Summary

Understanding these set operations is essential for working with sets in various mathematical contexts. They help in analyzing relationships between groups of elements and are the foundation for more advanced concepts.