Number - Set theory and Venn diagrams

Sets and Elements: A set is a well-defined collection of distinct objects called elements. We use the symbol $\in$ to denote that an object is an element of a set, such as $x \in A$. Visually, a set is represented by a closed loop (usually a circle) with its elements placed inside the boundary.

Universal Set and Complement: The Universal Set $\mathcal{U}$ contains all possible elements under consideration. In a Venn diagram, this is represented by a large bounding rectangle. The complement of set $A$, denoted as $A'$ or $A^c$, consists of all elements in $\mathcal{U}$ that are not in $A$. Visually, $A'$ is the entire region inside the rectangle but outside the circle $A$.

Intersection ($A \cap B$): The intersection of two sets $A$ and $B$ contains only the elements that belong to both sets. In a Venn diagram, this is represented by the 'overlapping' middle section where the two circles cross each other.

Union ($A \cup B$): The union of sets $A$ and $B$ contains all elements that are in $A$, or in $B$, or in both. Visually, this is the total area covered by both circles combined, including the intersection.

Subsets and the Empty Set: A set $A$ is a subset of $B$ ($A \subseteq B$) if every element of $A$ is also an element of $B$. Visually, this is shown by drawing circle $A$ entirely inside the boundary of circle $B$. The empty set, $\emptyset$ or $\{\}$, contains no elements and is a subset of every set.

Cardinality and Regions: Cardinality, denoted $n(A)$, refers to the number of elements in set $A$. In Venn diagrams, numbers written in regions typically represent the count of elements in that specific partitioned area. To find the total $n(A)$, you must sum the values in all regions contained within the circle $A$.

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