In previous posts we talked about how to represent a set algebraically. In this post, we will show it visually through the Venn Euler Diagram.

Venn-Euler Diagram Representation

In this type of diagram representation, we use a circle or rectangle, which will depend on the case. For example, the sets:

Example 1: Let the set $A = \\{\text{províncias do Japão}\\}$ be represented in a Venn-Euler Diagram:

Example 1.

Example 2: Let the set $B = \\{3,4,5,6\\}$ be represented in a Venn-Euler Diagram:

Example 1.

Intersection of Sets

In some cases, we have common elements between two or more sets. These cases are treated with the intersection between these sets.

Notation: We will use the symbol $\cap$ to represent the operation of intersection between two sets.

Example 1.

Thus, in algebraic terms, we represent:

$$X \cap Y = \\{a; ~ a \in X \text{ e } a \in Y \\}.$$

Example 3: Let the sets $A = \\{1,3,5\\}$ and $B = \\{5,6,7\\}$ be the representation in a Venn-Euler Diagram:

Example 1. $$A\cap B = \\{ 5 \\}$$

Empty Intersection

Definition (Empty Intersection): when two sets $A$ and $B$ have no elements in common.

$$A\cap B = \emptyset$$

We say we have Empty Intersection.

Example 4: Let the sets $A = \\{1,3,5\\}$ and $B = \\{2,4,6\\}$ be the representation in a Venn-Euler Diagram:

Example 1. $$A\cap B = \emptyset$$

Definition (Disjoint Sets): in the case of two sets with empty intersection they are said to be disjoint sets.

Chained Intersection

Definition (Linked Intersection): when the sets $A$ and $B$ are such that all elements of $A$ are also elements of $B$.

$$A\cap B = A\quad$$

because $A$ is contained in $B$, symbolically $A\subseteq B$. In this case, we say we have Chained Intersection.

Example 5: Let the sets $A = \\{1,3,5\\}$ and $B = \\{1,2, 3,4, 5, 6\\}$ be the representation in a Venn-Euler Diagram:

Example 1. $$A\cap B = A = \\{1,3,5\\}$$
  • In this example, we say that $A$ is a subset of $B$ or that A is contained in $B.$ and we use the symbology $A \subseteq B.$
  • We can also say that $B$ contains $A,$ and we write $B \supseteq A.$
  • As an important particular case, note that we always have $A \subseteq A.$

Note: when a set is not contained, we denote it with the symbol.

$$B \cancel{\subseteq} A$$

These are the concepts for introducing the Venn diagram. Wait for the next posts.