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:

How is the set of natural numbers written?

• $\N$ - the set of natural numbers.

Written as

What does $\Z$ represent?

• $\Z$ - the set of integers.

Written as

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

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

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

to denote the same relationship:

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."

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