Category Theory for the Sciences


What is a set?

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

How do you denote an Empty Set?

What does the assignment operator :=:= mean?

What does R\mathbb R designate?

Written as:

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

How is the set of natural numbers written?

Written as

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

What does Z\Z represent?

Written as

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

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

What is the symbol for exists?

What is the symbol for there exists a unique?

What does \forall denote in set builder notation?

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

The set of even integers can be written as

{nZ n is even}\{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 22 can be written as

{nZ n>2} or {nN n>2} or {nN n3}\{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:

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

to denote the same relationship:

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

How do you read the statement !xR such that x2=0\exists! x \in \mathbb R \text{ such that } x^{2} = 0

"There is one and only one number whose square is 00".

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

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

Exercises Let A={1,2,3}A = \{1, 2, 3\}.

What are all the subsets of AA? Hint: there are 88.

  1. A\emptyset \subseteq A

  2. AAA \subseteq A

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

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

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

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

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

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