mathematics-physics-wiki/docs/en/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.

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.

??? 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 $B$.