# Category Theory for the Sciences

## Sets

What is a set?

A set $X$ can be thought of as a collection of things $x \in X$. If there exists a pair $x, x' \in X$, one can tell if $x = x'$ or not.

How do you denote an Empty Set?

• $\emptyset$ - a set with no elements

What does the assignment operator $:=$ mean?

• $:=$ - define "this" to be "that".

What does $\mathbb R$ designate?

• $\mathbb R$ - the set of all real numbers.

Written as:

$\mathbb R := \{-1, \pi, \sqrt{45}, 1.8, 500, ...\}$

How is the set of natural numbers written?

• $\N$ - the set of natural numbers.

Written as

$\N := \{0, 1, 2, 3, 4, ..., 877, ...\}$

What does $\Z$ represent?

• $\Z$ - the set of integers.

Written as

$\Z := \{..., -551, ..., -2, -1, 0, 1, 2, ...\}$

When you want to say that some set is part of another set, what is that called? How is that denoted?

• $\subseteq$ - is a subset of the set.

Example, given that $\N$, contains only natural numbers whereas $\Z$ contains all integers, we can say that $\N \subseteq \Z$

What is the symbol for exists?

• $\exists$ - there exists.

What is the symbol for there exists a unique?

• $\exists!$ - there exists a unique.

What does $\forall$ denote in set builder notation?

• $\forall$ - for all.

How can you use set notation to write the set of even integers?

The set of even integers can be written as

$\{n \in \Z \ | \ n\ \text{is even}\}$

How can you use set notation to write the set of integers greater than 2?

The set of integers greater than $2$ can be written as

$\{n \in \Z \ | \ n > 2 \} \text{ or } \{n \in \N \ | \ n > 2 \} \text{ or } \{n \in \N \ | \ n \geq 3 \}$

using the symbol, $\exists$, one could also write:

$\{n \in \Z \ | \exists m \in \Z \text{ such that } 2m = n\}$

to denote the same relationship:

$\{n \in \Z \ | \ n\ \text{is even}\}$

How do you read the statement $\exists! x \in \mathbb R \text{ such that } x^{2} = 0$

"There is one and only one number whose square is $0$".

How do you use set builder notation to write the expression, "for all $m$ in $\N$, there exists $n$ in $\N$ such that for every number, there is a bigger one."

$\forall m \in \N \ \exists n \in \N \text{ such that } m < n$

### Exercises

2.1.1.2 Let $A = \{1, 2, 3\}$.

What are all the subsets of $A$? Hint: there are $8$.

1. $\emptyset \subseteq A$

2. $A \subseteq A$

3. $\{1\} \subseteq A$

4. $\{2\} \subseteq A$

5. $\{3\} \subseteq A$

6. $\{1, 2\} \subseteq A$

7. $\{2, 3\} \subseteq A$

8. $\{1, 3\} \subseteq A$