Discrete Mathematics #02 | Set Equality
In the previous lecture, we talked about sets as structure. This article will go a step further and talk about Venn diagrams and a simple set relationship which is set equality. In other words, when do two sets are equal. To make the equality of sets a reality, we must speak of the Axiom of Εxtensions and Venn Diagrams.
Venn diagrams
A Venn diagram is an iconographic representation of sets, popularized by John Venn in the 1880s. It is a diagram that shows all possible logical relations between a finite collection of different sets. Let's assume that we have a set A that contains the letters {a,b}. The Venn diagram will be:
The union of two sets A and B is denoted by A∪B. It results in a set containing all of A and B members. For example, suppose A = {1, 2, 3} and B = {2, 3, 4} and then A∪B = {1, 2, 3, 4}. The union of sets is a commutative operation: A∪B = B∪A. Venn is used to depicting these acts iconographically.
The intersection of two sets A and B is denoted by A ∩ B. It results in a set
containing the elements A and B have in common. Suppose A = {1, 2, 3} and B = {2, 3, 4} then A ∩ B = {2, 3}. Set intersection is a commutative operation: i.e. A ∩ B = B ∩ A.
The axiom of extensionality, or axiom of extension
First of all, we need to tackle the question "What is an axiom". It's a hypothesis that is taken to be true. In other words, a self-evident rule. So, the axiom says:
Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.
To simplify the rule, we will paraphrase it. A set A is equal to a set B if and only if every element of A exists in Β and every element of B exists in A.
For example, suppose A = {1, 3, 4} and B = {1, 3, 3, 4}. Are these sets equal (A = B)? Let's apply the axiom to these sets to find out! The number 1 is present in both sets. The same goes for number 4. What about 3? The number three is present in both sets, but B has a duplicate. Sets do not allow duplicates, and therefore, we only count one of them. This means that A is equal to B.
Notation
Venn diagrams are an excellent visual representation of a set. But how do we represent them as a mathematical expression? We use the set-Builder notation, which is a translation of a sentence into a mathematical expression. For example, let's say we have a set A = {1, 2, 3, 4}. Our friend Evangelos says, give me a mathematical expression of a new set, called B, such that every number of set A is greater than 1. Let's do that:
Step 1: Create your new set ("called B")
B = {}
Step 2: Translate the part of the sentence "every number of set A". To do that, we need to introduce the notation belongs ∈ and every ∀:
B = {∀x ∈ A}
Step 3: "Such that". In the sets theory, the "such that" phrase is converted into the symbol |. Therefore:
B = {∀x ∈ A| }
Step 3: "are greater than 1". This means that after the "such that" phrase, we need to put a condition in order to fulfil the sentence requirement.
B = {∀x ∈ A| x > 1}
Voila! Now, you can hand over our mathematical expression to our friend Evangelos. If we ask you more, you can read the table below and create even more complex formulas.
Conclusion
That was the end of today's lesson. In the next one, we will discuss the Set Theoretical Operations like union and intersection in more detail!
