# Chapter 2: Naive Set Theory

As our logic syntax we use the symbols $\forall$, $\exists$, $\vee$, $\wedge$, $!$, $\neg$, $\Rightarrow$ and $\Leftrightarrow$ to denote 'for all', 'exists', 'or', 'and', 'unique', 'not', 'implies' and 'if and only if' (or 'iff', meaning implies and implied) respectively.

We typically define a new notion by saying that something is called **'name'** if it satisfies a certain condition. In definitions, this 'if' is meant as an 'if and only if' in that we will call something 'name' if and only if it satisfies the condition of the definition.

## 2.1 Sets and functions

> **Slogan.** Sets are the machine code of modern Mathematics.

On a fundamental level, modern Math is built on Set Theory. From that point of view, a set $S$ is a collection of elements such that for every object $x$ in our 'universe' we can determine whether $x$ is an element of $S$, denoted $x \in S$ or that $x$ is not an element of $S$, denoted $x \notin S$.

When we want to specify the elements of a set $S$, we do so with bounding curly brackets and commas separating between elements e.g. $S = \{a, b, c\}$. Repeated elements in a set are ignored so $\{1, 1, 2, 3\} = \{1, 2, 3\}$. Also, the order of elements does not matter, so $\{2, 3, 1\} = \{1, 2, 3\}$.

If $S$ has finite number of elements (or just 'finite') we denote by $\#S$ (or $|S|$) the number of elements of $S$. Of course, $S$ need not be finite, and in this case, we need a rule in order to specify the elements of $S$, e.g. $S = \{n \mid n \text{ is a natural number and } n \geq 2\}$ or if the rule is clear after a few cases, we can write $S = \{2, 3, 4, \dots\}$. For sets $A, B$ we write $A \subseteq B$ if $\forall a \in A$ we have $a \in B$ and say that $A$ is included in $B$. Observe that for sets $A, B$, $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$. The basic operations on sets include

**union**
$$A \cup B = \{x \mid x \in A \vee x \in B\},$$

**intersection**
$$A \cap B = \{x \mid x \in A \wedge x \in B\},$$

and **complement** (or subtraction)
$$A \setminus B = \{x \mid x \in A \wedge x \notin B\}.$$

> **Remark 2.1.** More generally, let $I$ be a set that we refer to as an 'index set'. Suppose that for every $i \in I$ we are given a set $U_i$. Then we can form the union
> $$\bigcup_{i \in I} U_i = \{x \mid \exists i \in I : x \in U_i\}$$
> and the intersection
> $$\bigcap_{i \in I} U_i = \{x \mid \forall i \in I : x \in U_i\}.$$

Our fundamental assumption is that there exist a special set, called the **empty set** and denoted $\emptyset$ that has no elements. More formally, we can write
$$\emptyset = \{x \mid x \neq x\}$$
and observe that for every set $A$ we have $\emptyset \subseteq A$. Using the empty set, we can in fact define all natural numbers as follows:

$$0 := \emptyset,$$
$$1 := \{\emptyset\},$$