A set can be specified by enumeration: simply listing out its elements.
Set comprehension
By the axiom of specification, a set can be specified by filtering the elements of a larger set by some logical formula.
Subset of a set
Proper subset of a set
Equality of sets
An equivalence relation between sets. \(A\) and \(B\) are equal sets if they contain the same elements. \(A = B\) if when \({x \in{} A}\), \({x \in{} B}\) and when \({x \notin{} A}\), \({x \notin{} B}\).
Applying the contrapositive to the above, we get an equivalent definition: \[{x \in{} A \iff{} x \in{} B}\] This definition can be recogized as a statement about subsets: \[{A = B \iff{} A \subset{} B \text{and} B \subset{} A}\]
Cardinality of a set
Finite set
Infinite set
Countable set
Countably infinite set
Uncountable set
Empty set
The empty set is the unique set with no elements. It is often denoted by \({{\left\{\right\}}}\), \(\emptyset\) or \(\varnothing\).
Union of \(A\) and \(B\)
\({x \in{} A \cup{} B}\) if \({x \in{} A}\) or \({x \in{} B}\).
Intersection of \(A\) and \(B\)
\({x \in{} A \cap{} B}\) if \({x \in{} A}\) and \({x \in{} B}\).
Complement of \(A\) in \(\Omega\)
\({x \in{} {A}^{C}}\) if \({x \notin{} A}\).
Difference of \(A\) and \(B\)
\({x \in{} A \setminus{} B}\) if \({x \in{} A}\) and \({x \notin{} B}\).
Symmetric difference of \(A\) and \(B\)
Cartesian product of \(A\) and \(B\)
Power set of a \(A\)
The set of all possible subsets of \(A\). If \(A\) is finite, the cardinality of \({{\mathcal{P}} (A)}\) is \({2}^{{{{\lvert{} A \rvert}}}}\)
Equality by subset
If \({A \subset{} B}\) and \({B \subset{} A}\) then \(A\) and \(B\) are equal.
The empty set is the unique set with no elements. It is often denoted by \({{\left\{\right\}}}\), \(\emptyset\) or \(\varnothing\).
Set of natural numbers
Set of integers
Set of rational numbers
Set of real numbers
Can be constructed as the limits of all Cauchy sequences of rationals.
Set of irrational numbers
Set of complex numbers
Can be constructed from completing the real numbers.
Set of sets
Set of matrices
Relation
Binary relation
Reflexive relation
Symmetric relation
Transitive relation
Antisymmetric relation
Equivalence relation
Order
Ordered set
Partial order
Poset
Sequence of sets
Union of \({A}_{{1:\infty}}\)
\({x \in{} {\cup}_{n} {A}_{n}}\) if \({{\exists}_{n} x \in{} {A}_{n}}\)
Intersection of a sequence of sets
\({x \in{} {\cup}_{n} {A}_{n}}\) if \({{\forall}_{n} x \in{} {A}_{n}}\)
De Morgan’s law for sequences
Proof
Increasing sequence of sets
Strictly increasing sequence of sets
Decreasing sequence of sets
Strictly decreasing sequence of sets
Monotone sequence of sets
Strictly monotone sequence of sets
Limit supremum of a sequence of sets
The limit supremum is the set of points that reocurr. Possibly these points are cyclic or oscillating. \[{x \in{} {{\operatorname{LimSup} A}} \text{ if } \forall{} n \exists{} m \ge{} n, x \in{} {A}_{m}}\] \[{x \in{} {{\operatorname{LimSup} A}} \text{ if } x \in{} {{{\bigcap}}_{n}}^{{1:\infty}} {{{\bigcup}}_{m}}^{{n:\infty}} {A}_{m}}\] \[{x \in{} {{\operatorname{LimSup} A}} \text{ if } x \in{} {{\underset{{n \to{} \infty}}{\operatorname{Lim}} {{{{\bigcup}}_{m}}^{{n:\infty}} {A}_{m}}}}}\] We can think of the limit supremum as starting with the supremum, then as \(n\) grows, the points that will never reocurr again are gradually removed.
Limit infimum of a sequence of sets
The limit infimum is the set of points that reocurr in every step. \[{x \in{} {{\operatorname{LimInf} A}} \text{ if } \exists{} n \forall{} m \ge{} n, x \in{} {A}_{m}}\] Since the inner intersection is an increasing sequence, the outer union is simply there to let us take the limit. \[{x \in{} {{\operatorname{LimInf} A}} \text{ if } x \in{} {{{\bigcup}}_{n}}^{{1:\infty}} {{{\bigcap}}_{m}}^{{n:\infty}} {A}_{m}}\] Note that the inner intersection is monotone, therefore by a result in this chapter, the limit exists. Therefore the limit infimum may be written as follows, and it may be easier to think about it like this: \[{x \in{} {{\operatorname{LimSup} A}} \text{ if } x \in{} {{\underset{{n \to{} \infty}}{\operatorname{Lim}} {{{{\bigcup}}_{m}}^{{n:\infty}} {A}_{m}}}}}\] Also, we can think of the limit infimum as starting with the infimum, then as \(n\) grows, gradually adding on the points that eventually always reocurr.
Limit of a sequence of sets
The limit of a sequence of sets is defined to be equal to the limit supremum and the limit infimum of the sequence. It only exists if the limit supremum and limit infimum are equal.
Note that when computing the bound of an interval, the type of the resulting bound (open or closed boundary) can depend on whether the bounds are shrinking or expanding the set as \({n \to{} \infty}\). Shrinking will cause a closed bound, while expansion will cause an open bound.
Example
Example of computing the limit of a sequence of sets.
Monotone infimum is limit infimum
Let \(A\) be a decreasing sequence. \[{{{\operatorname{LimInf} A}} = {{\operatorname{Inf} A}}}\] Let \(A\) be an increasing sequence. \[{{{\operatorname{LimSup} A}} = {{\operatorname{Sup} A}}}\]
Proof
Limit of monotone sequence
The limit of a monotone sequence exists. Let \(A\) be a monotone sequence. \[{{{\operatorname{Lim} A}} = {{\operatorname{LimInf} A}} = {{\operatorname{LimSup} A}}}\]