Sets – Steven Golovkine

This post is an introduction to sets as mathematical objects. It is based on the book Introduction to Topology of Menselson [1]. It aims to give a visual introduction to sets theory.

Roughly speaking, a set $S$ is an ensemble of objects. Let $x$ be an object. If $x$ belongs to $S$, we will note $x \in S$ and if $x$ does not belong to $S$, we will note $x \notin S$. We will also note $\{x_1, \dots, x_n\}$ the set that contains the objects $x_1, \dots, x_n$. The set that has no member is called to null (or empty) set and noted $\emptyset$.

When all objects of a set $A$ also belongs to a set $B$, we say that $A$ is a subset of $B$ and we note $A \subset B$. Note that a set is always contained in itself and the same is true for the empty set $\emptyset$, these are improper subsets. Others subsets are called proper subsets.

To compare multiple sets, we compare their elements. To show the equality of sets, it is usually a good idea to prove that each set is contained in each other.

Sets are also objects.

Power sets

The power set of $S$, noted $\mathcal{P}(S)$ or $2^S$, is a set that is the collection of all subsets of $S$. For each set $S$, the set $A \in \mathcal{P}(S)$ if and only if $A \subset S$.

Some comments about sets:

For each set $S$, $S \in \mathcal{P}(S)$ by definition but $S \not\subset \mathcal{P}(S)$ as the elements of $S$ do not belong to $\mathcal{P}(S)$. However, the set $\{S\} \subset \mathcal{P}(S)$ because all the elements of $\{S\}$, $S$ and $\emptyset$, are in $\mathcal{P}(S)$.
For each set $S$, $\emptyset \in \mathcal{P}(S)$ by definition and $\emptyset \subset \mathcal{P}(S)$ as the elements of $\emptyset$ (there are none) are included in $\mathcal{P}(S)$.
Consider $A, B$ and $C$. If $A \subset B$ and $B \subset C$, then $B \subset C$.

Intersection, Union and Complement

Let $A$ and $B$ be two sets. The set that contains elements that belongs to $A$ and to $B$ is the intersection of $A$ and $B$ and is denoted by $A \cap B$. The set that contains elements that belongs to $A$ or $B$ is the union of $A$ and $B$ and is denoted by $A \cup B$.

The complement of a set $A \subset S$ is the set of elements that belong to $S$ but not to $A$. It is denoted by $C(A)$. As $C(A)$ is a set, we can take its complement and recover $A$.

DeMorgan’s laws

DeMorgan’s laws are relations between the complement of the intersection of $A$ and $B$ (or the union) and the union of the complement of $A$ and the complement of $B$ (or the intersection of the complements). Mathematically, this is written, for $A$ and $B$ two sets:

$C(A \cap B) = C(A) \cup C(B)$

Figure 7: Visual proof for the complement of the intersection

$C(A \cup B) = C(A) \cap C(B)$

Figure 8: Visual proof for the complement of the union

References:

[1] Mendelson, B., 2012. Introduction to Topology: Third Edition. Courier Corporation.

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--- title: Sets author: Steven Golovkine date: '2023-11-11' categories: ['Topology'] description: | A visual introduction to sets. image: "../../posts/sets/logo.jpg" image-alt: | A square logo of sets created with DALL-E3. --- This post is an introduction to sets as mathematical objects. It is based on the book *Introduction to Topology* of Menselson [1]. It aims to give a visual introduction to sets theory. Roughly speaking, a set $S$ is an ensemble of objects. Let $x$ be an object. If $x$ belongs to $S$, we will note $x \in S$ and if $x$ does not belong to $S$, we will note $x \notin S$. We will also note $\{x_1, \dots, x_n\}$ the set that contains the objects $x_1, \dots, x_n$. The set that has no member is called to *null* (or *empty*) set and noted $\emptyset$. ::: {#fig-sets fit-align="center"} ![](sets_figure_1.png){width=200} Sets ::: When all objects of a set $A$ also belongs to a set $B$, we say that $A$ is a subset of $B$ and we note $A \subset B$. Note that a set is always contained in itself and the same is true for the empty set $\emptyset$, these are *improper* subsets. Others subsets are called *proper* subsets. ::: {#fig-inclusion fit-align="center"} ![](sets_figure_2.png){width=200} Sets inclusion ::: To compare multiple sets, we compare their elements. To show the equality of sets, it is usually a good idea to prove that each set is contained in each other. ::: {.callout-important appearance="simple"} **Sets are also objects.** ::: ## Power sets The power set of $S$, noted $\mathcal{P}(S)$ or $2^S$, is a set that is the collection of all subsets of $S$. For each set $S$, the set $A \in \mathcal{P}(S)$ if and only if $A \subset S$. ::: {#fig-power-sets fit-align="center"} ![](sets_figure_3.png){width=200} Power sets ::: Some comments about sets: - For each set $S$, $S \in \mathcal{P}(S)$ by definition but $S \not\subset \mathcal{P}(S)$ as the elements of $S$ do not belong to $\mathcal{P}(S)$. However, the set $\{S\} \subset \mathcal{P}(S)$ because all the elements of $\{S\}$, $S$ and $\emptyset$, are in $\mathcal{P}(S)$. - For each set $S$, $\emptyset \in \mathcal{P}(S)$ by definition and $\emptyset \subset \mathcal{P}(S)$ as the elements of $\emptyset$ (there are none) are included in $\mathcal{P}(S)$. - Consider $A, B$ and $C$. If $A \subset B$ and $B \subset C$, then $B \subset C$. ::: {#fig-multiple-inclusion fit-align="center"} ![](sets_figure_4.png){width=200} Multiple sets inclusion ::: ## Intersection, Union and Complement Let $A$ and $B$ be two sets. The set that contains elements that belongs to $A$ **and** to $B$ is the intersection of $A$ and $B$ and is denoted by $A \cap B$. The set that contains elements that belongs to $A$ **or** $B$ is the union of $A$ and $B$ and is denoted by $A \cup B$. ::: {#fig-inter-union layout-ncol=2} ![Intersection](sets_figure_5.png){width=175} ![Union](sets_figure_6.png){width=200} Intersection and Union ::: The complement of a set $A \subset S$ is the set of elements that belong to $S$ but not to $A$. It is denoted by $C(A)$. As $C(A)$ is a set, we can take its complement and recover $A$. ::: {#fig-complements fit-align="center"} ![](sets_figure_7.png){width=200} Complement ::: ## DeMorgan's laws DeMorgan's laws are relations between the complement of the intersection of $A$ and $B$ (or the union) and the union of the complement of $A$ and the complement of $B$ (or the intersection of the complements). Mathematically, this is written, for $A$ and $B$ two sets: - $C(A \cap B) = C(A) \cup C(B)$ ::: {#fig-visual-inter fit-align="center"} ![](sets_figure_8.png){width=400} Visual proof for the complement of the intersection ::: - $C(A \cup B) = C(A) \cap C(B)$ ::: {#fig-visual-union fit-align="center"} ![](sets_figure_9.png){width=400} Visual proof for the complement of the union ::: References: [1] Mendelson, B., 2012. Introduction to Topology: Third Edition. Courier Corporation.