Venn Euler diagrams are useful for a visual representation of sets. In OBMEP 2022 a question appeared, using this theory. In this post, we will describe operations with these diagrams.

Euclid's work.

Union of sets

Union of sets is one of the most used operations on sets. It will be the starting point for operations with sets, defined by:

Definition (Union of sets) Given two sets $A$ and $B,$ the union of these sets is formed by all the elements that belong to $A$ or $B$ ($A\cup B$). In algebraic terms, we can write:

$$A\cup B = \\{x;~x\in A ~\text{ ou } x\in B\\}$$
Euclid's work.

Cases in Union of Sets

Disjoint Sets Sets with Intersection Chained Sets
Example 1. Example 1. Example 1.

Intersection of sets

Set Intersection is another operation, well used on sets. It is defined by:

Definition (Intersection of sets) Given two sets $A$ and $B,$ the intersection of these sets is formed by all the elements that belong to $A$ and $B$ at the same time ($A\cap B$). In algebraic terms, we can write:

$$A\cap B = \\{x;~x\in A ~\text{ e } x\in B\\}$$
Euclid's work.

Cases in Set Intersection

Empty Intersection $(A\cap B = \emptyset = \\{~~\\})$ Sets with Intersection $(A\cap B = C)$ Subsets $(A\cap B = B)$
Example 1. Example 1. Example 1.

Intersection Application

(OBMEP 2022 - Level 1 - Question 6) The first phase of OBMEP consists of three tests, at levels 1, 2 and 3 (N1, N2, N3), with 20 questions in each test.

In these tests:

  • three questions are common to the three levels
  • two are common only to levels 1 and 2 and
  • two others are common only to levels 2 and 3.
  • the other questions only appear in one of the tests.

How many different questions appear in the three tests in the first phase of OBMEP?

Euclid's work.

Set difference

Set Difference is a very common use in this theory. It is defined by:

Definition (Difference of sets) Given two sets $A$ and $B,$ the difference of these sets is formed by all the elements that belong to $A,$ but do not belong to $B$ ($A- B$). In algebraic terms, we can write:

$$A - B = \\{x\in A \text{ tal que } x\notin B\\}$$
Euclid's work.

Cases in Set Difference

Empty Difference $(A - B = \emptyset = \\{~~\\})$ Difference with Intersection (Pac-man) Donut
Example 1. Example 1. Example 1.
Note: The difference with partial intersection of sets has a format similar to pac-man.
Euclid's work.
Note: The difference with total intersection of sets has a donut-like shape.
Euclid's work.

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