> *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.
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> *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 $\mathbb{N} \subsetneq \mathbb{Z}$.
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 $B$.
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 $B$.
