5.4 KiB
Sets
Sets and subsets
Definition: a set is a collection of elements uniquely defined by these elements.
Examples are \mathbb{N}
, the set of natural numbers. \mathbb{Z}
, the set of integers. \mathbb{Q}
, the set of rational numbers. \mathbb{R}
, the set of real numbers and \mathbb{C}
the set of complex numbers.
Definition: suppose
A
andB
are sets. ThenA
is called a subset ofB
, if for every elementa \in A
there also isa \in B
. ThenB
containsA
and can be denoted byA \subseteq B
.
The extra line under the symbol implies properness. A subset A
of a set B
which is not the empty set \varnothing
nor the full set B
is called a proper subset of B
, denoted by A \subsetneq B
. For example \mathbb{N} \subsetneq \mathbb{Z}
.
Definition: if
B
is a set, then\wp(B)
denotes the set of all subsetsA
ofB
. The set\wp(B)
is called the power set ofB
.
Suppose for example that B = {x,y,z}
, then \wp(B) = \{\varnothing,\{x\},\{y\},\{z\},\{x,y\},\{x,z\},\{y,z\},\{x,y,z\}\}
.
Proposition: let
B
be a set withn
elements. Then its power set\wp(B)
containsw^n
elements.
??? note "Proof:"
Let $B$ be set with $n$ elements. A subset $A$ of $B$ is completely determined by its elements. For each element $b \in B$ there are two options, it is in $A$ or it is not. So, there are $2^n$ options and thus $2^n$ different subsets $A$ of $B$.
Proposition: suppose
A
,B
andC
are sets. Then the following hold:
- if
A \subseteq B
andB \subseteq C
thenA \subseteq C
,- if
A \subseteq B
andB \subseteq A
thenA = B
.
??? note "Proof:"
To prove 1, suppose that $A \subseteq B$. Let $a \in A$, then $a \in B$ therefore $a \in C$.
To prove 2, every element of $A$ is in $B$ and every element of $B$ is in $A$. As the set is uniquely determined by its elements $A = B$.
Definition: let
P
be a predicate with reference setX
, then
\big{x \in X ;\big|; P(x) \big}
denotes the subset of
X
consisting of all elementsx \in X
for which statementP(x)
is true.
Operations on sets
Definition: let
A
andB
be sets.
- The intersection of
A
andB
(A \cap B)
is the set of all elements contained in bothA
andB
.- The union of
A
andB
(A \cup B)
is the set of elements that are in at least on ofA
orB
.A
andB
are disjoint if the intersection(A \cap B)
is the empty set\varnothing
.
Definition: suppose
I
is a set (an index set) and for each elementi
there exists a setA_i
, then
\bigcup_{i \in I} A_i := \big{x ;\big|; \text{there is an } i \in I \text{ with } x \in A_i \big},
and
\bigcap_{i \in I} A_i := \big{x ;\big|; \text{for all } i \in I \text{ there is } x \in A_i \big}.
Implying unions and intersections taken over an index set. For example suppose for each i \in \mathbb{N}
the set A_i
is defined as \{x \in \mathbb{R} \;|\; 0 \leq x \leq i \}
, then
\bigcap_{i \in \mathbb{N}} A_i = {0},
and
\bigcup_{i \in \mathbb{N}} A_i = \mathbb{R}_{\geq 0}.
Definition: if
C
is a collection of sets, then
\bigcup_{A \in C} A := \big{x ;\big|; \text{there is an } A \in C \text{ with } x \in A \big},
and
\bigcap_{A \in C} A := \big{x ;\big|; \text{for all } A \in C \text{ there is } x \in A \big}.
- Definition: let
A
andB
be sets. The difference ofA
andB
(A \backslash B)
is the set of all elements fromA
that are not inB
.- The symmetric difference of
A
andB
(A \triangle B)
is the set consisting of all elements that are in exactly one ofA
orB
.If one is working inside a fixed set
U
and only considering subsets ofU
, then the differenceU \backslash A
is also called the complement ofA
inU
, denoted byA^*
. In this case the setU
is called the universe.
Cartesian products
Suppose a_1, a_2, \dots, a_k
are elements from some set, then the ordered k-tuple of a_1, a_2, \dots, a_k
is denoted by (a_1, a_2, \dots, a_k)
- Definition: the cartesian product
A_1 \times \dots \times A_k
of setsA_1, \dots , A_k
is the set of all ordered k-tuples(a_1, a_2, \dots, a_k)
wherea_i \in A_i
for1 \leq i \leq k
.- If
A
andB
are sets then
A \times B = \big{ (a,b) ;\big|; a \in A,; b \in B \big}
Notice that for all 1 \leq i \leq k
and A_i = A
then A_1 \times \dots \times A_k
is also denoted by A^k
.
Partitions
Definition: let
S
be a nonempty set. A collection\Pi
of subsets is called a partition if and only if
\varnothing \notin \Pi
,\bigcup_{X \in \Pi} X = S
,- for all
X \neq Y \in \Pi
there isX \cap Y = \varnothing
For example the set \{1,2, \dots , 10\}
can be partitioned into the sets \{1,2,3\}
, \{4,5\}
and \{6,7,8,9,10\}
.
Quantifiers
Definitions: the universal quantifier "for all" is denoted by
\forall
and the existential quantifier "there exists" is denoted by\exists
.
Proposition - DeMorgan's rule: the statement
\neg (\forall x \in X ;[P(x)])
is equivalent with the statement
\exists x \in X ;[\neg (P(x))].
The statement
\neg (\exists x \in X ;[P(x)])
is equivalent with the statement
\forall x \in X ; [\neg (P(x))].
??? note "Proof:"
will be added later.