Sets - Venn Diagrams

Venn Diagrams are graphical representations of sets where a Universal Set $U$ is depicted as a large rectangle, and its subsets are shown as circles or closed curves within it. The position and overlap of these circles represent the relationships between the sets.

The Union of two sets $A \cup B$ is visually represented by shading the entire area covered by circle $A$ and circle $B$. This includes the overlapping region. It represents elements that belong to either $A$, $B$, or both.

The Intersection of two sets $A \cap B$ is shown as the common area where circle $A$ and circle $B$ overlap. If the circles do not overlap, the sets are called Disjoint Sets, and $A \cap B = \phi$.

The Complement of a set $A$, denoted as $A'$ or $A^c$, is the region inside the universal rectangle $U$ but outside the circle $A$. This represents all elements in the universal set that are not in set $A$.

The Difference of sets $A - B$ (also called the relative complement) is the region of circle $A$ that does not overlap with circle $B$. Visually, it looks like a crescent moon shape within circle $A$. It represents elements belonging to $A$ but not to $B$.

A Subset relationship $A \subset B$ is represented visually by placing circle $A$ entirely inside circle $B$. This indicates that every element in $A$ is also an element of $B$.

The Symmetric Difference $A \Delta B$, defined as $(A - B) \cup (B - A)$, is represented by shading the portions of circles $A$ and $B$ that do not overlap. The central intersection area remains unshaded.

De Morgan's Laws can be visualized using Venn Diagrams: $(A \cup B)'$ is the region outside both circles, which is identical to the intersection of the regions outside $A$ and outside $B$ ($A' \cap B'$).