Sets and Functions - Venn Diagrams and Operations on Sets

A Set is a well-defined collection of distinct objects. The Universal Set ($U$) represents the totality of all elements under consideration for a specific problem. Visually, $U$ is depicted as a large rectangle which acts as the boundary for all other sets.

The Union of two sets $A$ and $B$, denoted as $A \cup B$, is the set containing all elements that belong to $A$, or $B$, or both. In a Venn diagram, this is represented by shading the entire region covered by the two overlapping circles $A$ and $B$.

The Intersection of two sets $A$ and $B$, denoted as $A \cap B$, consists of elements that are common to both sets. Visually, this is the region where the circles for $A$ and $B$ overlap. If the circles do not overlap, the sets are disjoint and the intersection is the empty set $\emptyset$.

The Complement of a set $A$, written as $A'$ or $A^c$, includes all elements in the universal set $U$ that are not in $A$. Visually, this corresponds to the entire area inside the universal rectangle excluding the interior of circle $A$.

The Difference of sets $A - B$ (also called the relative complement) consists of elements that belong to $A$ but not to $B$. On a Venn diagram, this is shown as the 'crescent' part of circle $A$ that does not share any area with circle $B$.

The Symmetric Difference of sets $A$ and $B$, denoted as $A \Delta B$, is the set of elements belonging to either $A$ or $B$ but not both. Visually, it is the union of the two circles excluding their shared overlapping intersection region.

Disjoint Sets are sets that have no elements in common, meaning $A \cap B = \emptyset$. Visually, they are represented as two separate circles within the universal rectangle that do not touch or intersect.

A Subset relationship, $A \subseteq B$, occurs when every element of set $A$ is also an element of set $B$. Visually, circle $A$ is drawn entirely inside the boundary of circle $B$.