1 - Set Theory

Homework 1. Reading 1.

What are sets?

A set is a collection of objects called elements. You can list a set like this:

$S=\{1,2,3\}$

$5\in S$ means that 5 belongs to set S. You can also write $1,2,3\in S$.

x $\notin$ y means that x is not a set of y

The order of a set does not matter

Sets are made up of unique elements. If there are duplicates within a set, ignore them. Thus $\{1,2,3,1\}$ is the same set as $\{1,2,3\}$. The two sets are equal because they have the same elements.

Sets can have numbers, but also letters, symbols etc.

$\mathbb{N}$ is the set of natural numbers. (All nonzero positive integers)

$\mathbb{Z}$ is commonly used to represent the set of all integers.

$\mathbb{Q}$ = rational numbers

$\mathbb{R}$ = real numbers

$\mathbb{C}$ = complex numbers

You can write a set many different ways

For the set of even integers could be written as:

$\{...,-4,-2,0,2,4,\dots \}$

or as

$2x:x\in Z$

We read the colon as “such that,” so the above is read as “the set of elements of the form 2x such that x is an integer. Or 2x where x belongs to the natural set of integers. Writing sets with a colon is called set-builder notation

$\{x\in Z:x$ is an even integer} or {x : x=2y for some y ∈ Z}.

The Empty Set

The empty set is a set with no elements. It is written like this:

$\varnothing =\{\}$

Sometimes you want the empty set to be an element of another set:

$\{\{\}\}$ or $\{\varnothing \}$

You can think of it like a box with an empty box inside of it.

The empty set is a subset of any other set. But not an element of any other set.

Subsets

$A\subseteq B$ means that A is a subset of B. Add a slash to mean that A is not a subset of B.

To check that A is a subset of B, we check that every element of A also belongs to B.

or

To check that A is a subset of B, we check that we can form A by throwing out some of the elements of B.

Cardinality

Cardinality is the number of elements in a set

$|S|=3$ means that $S$ has a cardinality of 3.

$||$ can also mean absolute value. We determine this based off of context.

Power sets

A power set $S$, written is a set containing all of the subsets of $S$.

Example: $S=\{1,2,3\}$

Theorem: determine size of power set

Unions and Intersections

Union

A union is formed by combining two sets, for example $S$ and $T$. The union contains the set of elements which belong to S or T or both of them.

$S\cup T=\{x:x\in S$ or $x\in T\}$

Intersection

Intersections are sets formed from elements in both set $S$ and $T$.

$S\cap T=\{x:x\in S$ and $x\in T\}.$

Complements & differences

You do not need to worry about numbers not used in $S$

Homework 1

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