In previous posts we talked about how to represent, concepts of relevance and presented some important sets. Here, we are going to talk about equivalence and equality between two sets.

Equality of a set

Initially, let’s define the equality of two sets.

Definition (Equality of two sets): Two sets are equal if they have exactly the same elements.

Example 1.

Example 1: Let the sets $A$ and $B$ have the same elements:

$$A = \\{\text{inteiros múltiplos de } 2\\}$$

$$B = \\{2n; ~k\in \mathbb{Z}\\}$$

Notation: We write two equal sets $X$ and $Y$ as $X=Y.$

Different sets

Notation: We write two different sets $X$ and $Y$ as $X\neq Y.$

Example 1.

Example 2: Let the sets be $A$ and $B$:

  • $X=\\{1,2,3\\}$
  • $Y=\\{1,2\\}$
  • In this case $X\neq Y,$ because $X$ has an element ($3$), which does not belong to $Y$.

Equivalence of sets

Before talking about equivalence, we have to keep in mind the definition of one-to-one correspondence.

Definition (One-to-one Correspondence): is an expression given to sets in which they have the same number of elements in common.

Example 1.

With this, we can define the equivalence of two sets.

Definition (Equivalence of two sets): two sets are equivalent if their elements can be placed in one-to-one correspondence. More specifically, if we can associate each element of one of these sets with a single element of the other and vice versa. So, different elements from the first set are associated with different elements from the second set.

Example 1.
Note: note that two equal sets are equivalent, but two equivalent sets are not necessarily equal.

To understand the concept of set equivalence, look at the example:

Example 3: Let the sets be $A$ and $B$:

$$C = \\{a, b, c, d, e\\} \text{ e } D = \\{1, 2, 3, 4, 5\\}$$

are equivalent. Indeed, a possible one-to-one correspondence between its elements is:

  • $1 \leftrightarrow a$
  • $2 \leftrightarrow b$
  • $3 \leftrightarrow c$
  • $4 \leftrightarrow d$
  • $5 \leftrightarrow e$

In set notation $C \leftrightarrow D$.

Finite sets

The definition of equivalence between sets is fundamental to We define the following set:

Definition (Finite set): a set $A$ is finite if $A$ is empty or equivalent to a set of the form:

$$\\{1, 2, 3, \dots , n\\}$$

for some $n \in \mathbb{N}$ (belonging to the set of naturals).

Example 1.
Note: note if $A$ is empty ($A= \emptyset$), we say it has zero $(0)$ elements.

Cardinality of sets

Next, we will define the cardinality of a set.

Definition (Cardinality of a Set): Let $A$ be a finite set and equivalent to $\\{1, 2, 3, \dots , n\\}$. In this case we have that $A$ has $n$ elements or cardinality $n$.

Example 1.

The notation for this type of case is given below:

Notation: we denote the number of elements in a finite set $A$ by writing $|A|$ or $\\#A $

Example 3: Let the sets be $A$ and $B$, calculate the cardinality of the sets:

  • $A = \\{1,2,3\\}$
    • $\\#A = 3$
  • $B = \\{1,2,7,8\\}$
    • $\\#B = 4$
Example 1.
Note: Sets that are not finite are said to be infinite. Some examples of infinite sets are $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \\{x;~ x \text{ é inteiro par}\\}.$

These are the concepts for set equivalence. Wait for the next posts.