Sets - Operations on Sets: Union, Intersection, Difference

Union of Sets ($A \cup B$): The union of two sets $A$ and $B$ is the set containing all elements that belong to $A$, or to $B$, or to both. In a Venn diagram, this is visualized by shading the entire area of both overlapping circles.

Intersection of Sets ($A \cap B$): The intersection consists of only those elements that are common to both $A$ and $B$. On a Venn diagram, this is represented by the central football-shaped overlapping region where the two circles meet.

Disjoint Sets: Two sets $A$ and $B$ are said to be disjoint if their intersection is an empty set, i.e., $A \cap B = \emptyset$. Visually, these are represented as two separate circles that do not touch or overlap in the Venn diagram.

Difference of Sets ($A - B$): The difference $A - B$ is the set of elements that belong to $A$ but do not belong to $B$. Visually, it is shown as circle $A$ with the overlapping part shared with $B$ removed, appearing like a crescent moon shape.

Complement of a Set ($A'$): The complement of a set $A$ is the set of all elements in the Universal set $U$ that are not in $A$. In a Venn diagram, if $U$ is the bounding rectangle and $A$ is a circle inside it, $A'$ is the entire area inside the rectangle but outside the circle.

Symmetric Difference ($A \Delta B$): This operation is defined as $(A - B) \cup (B - A)$. It includes elements that are in either $A$ or $B$, but not in both. Visually, it shades the outer portions of both circles while leaving the middle intersection blank.

De Morgan's Laws: These are fundamental rules relating unions and intersections via complements: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$. These laws show how the complement of a union is the intersection of the complements, and vice versa.