Sets - Practical Problems on Union and Intersection of Two Sets

Cardinal Number of a Set: The number of distinct elements present in a finite set $A$ is called its cardinal number and is denoted by $n(A)$. For example, if $A = \{1, 2, 3, 4\}$, then $n(A) = 4$.

Union of Two Sets ($A \cup B$): This represents the collection of elements that belong to set $A$, or set $B$, or both. Visually, in a Venn diagram, this corresponds to the entire area enclosed by both overlapping circles representing $A$ and $B$.

Intersection of Two Sets ($A \cap B$): This represents elements that are common to both set $A$ and set $B$. In a Venn diagram, this is depicted as the overlapping region where the two circles cross each other.

Disjoint Sets: Two sets $A$ and $B$ are called disjoint if they have no common elements, meaning $A \cap B = \emptyset$. Visually, these are represented as two separate circles that do not touch or overlap within the universal set rectangle.

Difference of Sets ($A - B$ and $B - A$): The set $A - B$ contains elements present in $A$ but not in $B$. Visually, this is the region of circle $A$ that excludes the overlapping intersection part, resembling a crescent moon shape. Similarly, $B - A$ is the part of circle $B$ excluding the intersection.

The Inclusion-Exclusion Principle: To find the total elements in the union $n(A \cup B)$, we add $n(A)$ and $n(B)$, then subtract the intersection $n(A \cap B)$ because those elements were counted twice (once in $A$ and once in $B$).

Universal Set and Complements: The universal set $U$ is represented by a rectangle containing the circles. Elements that belong to neither $A$ nor $B$ are located inside the rectangle but outside both circles, denoted by $(A \cup B)'$.