Sets - Operations on Sets: Union and Intersection (using Venn diagrams)

Union of Sets ($A \cup B$): The union of two sets $A$ and $B$ is the set of all elements that belong to set $A$, or to set $B$, or to both. In a Venn diagram, this is represented by shading the entire region covered by both circle $A$ and circle $B$ within the universal set rectangle.

Intersection of Sets ($A \cap B$): The intersection of two sets $A$ and $B$ is the set of elements that are common to both $A$ and $B$. Visually, this corresponds to the overlapping middle region where circle $A$ and circle $B$ cross each other.

Disjoint Sets: Two sets are disjoint if they have no elements in common, meaning $A \cap B = \emptyset$. In a Venn diagram, disjoint sets are represented by two separate circles that do not touch or overlap at any point.

Overlapping Sets: Sets that have at least one common element are called overlapping sets. On a Venn diagram, these sets are drawn with an overlapping area in the center. The union $A \cup B$ includes the elements in the left circle, the right circle, and the overlap, while the intersection $A \cap B$ is only the overlap.

Universal Set ($\xi$ or $U$): The universal set contains all the elements under consideration for a particular discussion and is represented by a large rectangle. All other sets (represented by circles) are subsets of this rectangle.

Cardinal Number of Union: The number of elements in the union of two sets, denoted as $n(A \cup B)$, is found by adding the number of elements in $A$ and $B$, then subtracting the number of elements in their intersection to avoid double-counting. Visually, if you count all dots in circle $A$ and all dots in circle $B$, the dots in the overlapping area were counted twice and must be subtracted once.

Subsets in Venn Diagrams: If $B \subset A$, then every element of $B$ is also an element of $A$. In a Venn diagram, this is shown as circle $B$ being entirely contained inside circle $A$. In this case, $A \cup B = A$ and $A \cap B = B$.