notes/docs/mathematics/set-theory/sets.md

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 and B are sets. Then A is called a subset of B, if for every element a \in A there also is a \in B. Then B contains A and can be denoted by A \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 subsets A of B. The set \wp(B) is called the power set of B.

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 with n elements. Then its power set \wp(B) contains w^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 and C are sets. Then the following hold:

1. if A \subseteq B and B \subseteq C then A \subseteq C,
2. if A \subseteq B and B \subseteq A then A = 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 set X, then

\big{x \in X ;\big|; P(x) \big}

denotes the subset of X consisting of all elements x \in X for which statement P(x) is true.

Operations on sets

Definition: let A and B be sets.

• The intersection of A and B (A \cap B) is the set of all elements contained in both A and B.
• The union of A and B (A \cup B) is the set of elements that are in at least on of A or B.
• A and B 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 element i there exists a set A_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 and B be sets. The difference of A and B (A \backslash B) is the set of all elements from A that are not in B.

The symmetric difference of A and B (A \triangle B) is the set consisting of all elements that are in exactly one of A or B.

If one is working inside a fixed set U and only considering subsets of U, then the difference U \backslash A is also called the complement of A in U, denoted by A^*. In this case the set U 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 sets A_1, \dots , A_k is the set of all ordered k-tuples (a_1, a_2, \dots, a_k) where a_i \in A_i for 1 \leq i \leq k.

If A and B 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 is X \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.