Sets

Set: an unordered collection of objects.

$\in$ is used to indicate that an object belongs to a set.

$A\subseteq B$ A is a subset of B, each elements in A is also an element of B

$A=B$ two sets are equal if they contain exactly the same elements.

A convenient way to prove $A=B$ is:

1. prove that $A\subseteq B$, and
2. prove that $B\subseteq A$

Set Expressions

The “mathese” of set expression has the following form:

e.g. The set of non-negative numbers

$$\{x\in \mathbb{R}:x\ge 0\}$$

“The set consisting of all elements x of the set of real numbers such that x is greater or equal to 0”

Sometimes this gets simplified to:

$$\{x:x\ge 0\}$$

if it is considered clear what kind of values $x$ are supposed to take on.

Cardinality

If a set is not infinite $|S|$ denotes the number of elements, or the cardinality of the set.

Cartesian product

$A\times B$ is the set of all pairs $(a,b)$ where $a\in A$ and $b\in B$.

e.g.
for $A=\{1,2\}$ and $B=\{d,e,f,g\}$
$A\times B=\{(1,d),(1,e),(1,f),(1,g),(2,d),(2,e),(2,f),(2,g)\}$

if $A$ and $B$ are finite, then:

$$|A\times B|=|A|\times |B|$$

Tuples in set expressions

The set expression:

$$\{(x,y)\in \mathbb{R}\times \mathbb{R}:y=x^2\}$$

denotes the set of all pairs of real numbers in which the second element is the square of the first.

The set expression:

$$\{(x,y,z)\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}:x\ge 0,y\ge 0,z\ge 0\}$$

denotes the set of all triples consisting of non-negative numbers.
In the right-part of the expression, the commas have to be interpreted as “and” logical expression, so, a triple, to be included in the sets must satisfy all the listed condition on the right.