Equal set, subset, and power set

Naive set theory: Elementary notions and notations

Equal set, subset, and power set

Equality and inequality of sets

Equality / Inequality

Equality of two sets $A$ and $B$ , denoted as $A=B$ , means that they have the same elements. This means that every element of $A$ is also an element of $B$ and vice versa.

Inequality of two sets $A$ and $B$ , i.e., not being equal to each other, is denoted as $A\neq B$ .

Examples

$\{a,b,b\}=\{a,b\}=\{b,a\}$ $\{n\in\mathbb{N}\mid n^2<10\}$ ${}=\{n\in\mathbb{N}\mid n\le 3\}$ ${}=\{0,1,2,3\}\phantom{xxxxx}\!$ $\{a\}\neq\bigl\{\{a\}\bigr\}$

Equality of sets is also referred to in the literature as the axiom of extensionality in set theory.

Subset

Subset

A set $A$ is a subset of a set $B$ if every element of $A$ also belongs to $B$ . We denote this as $A\subset B$ . The notation $A\subseteq B$ is sometimes used to indicate that the sets can be equal. But where $\lt$ symbolizes "strictly less than" and $\le$ symbolizes "less than or equal to," the notations $\subset$ and $\subseteq$ refer to the same notion of inclusion, namely "is a subset of." If $A\subset B$ but $A\neq B$ , we denote this as $A\subsetneq B$ and call it a proper subset.

If $A$ is not a subset of $B$ , we denote this as $A\not\subset B$ .

Examples

$\mathbb{N}\subset\mathbb{Z}$ $\{1\}\subset\{1,2\}$ $\{1\}\subsetneq\{1,2\}$ $\{1,2\}\subset\{2,1\}$ $\{1,2\}\subsetneq\{n\in\mathbb{N}\mid n<4\}$ $\emptyset\subset\{1,2\}$ $\emptyset\subset\emptyset$ $\{a,b,c\}\not\subset\{a,b,d\}$

Symbols for inclusion Note: there are also textbooks in which $\subseteq$ symbolizes "is a subset of" and $\subset$ symbolizes "is a proper subset of." An example of this is Seymour Lipschutz, Set Theory, from the Schaum's series.

Superset If it is true that $A$ is a subset of $B$ , then we say that $B$ is a superset of $A$ , denoted as $B\supset A$ or $B\supseteq A$ .
If $B$ is a proper superset of $A$ , we can denote this as $B\supsetneq A$ .
If $B$ is not a subset of $B$ , we can denote this as $B\not\supset A$ .

Element vs subset

Do not confuse the symbol $\in$ with the symbol $\subset$ . For example, $\{1\}\in\bigl\{\{1\},2\bigr\}$ , but $\{1\}\not\subset\bigl\{\{1\},2\bigr\}$ because $1$ is not an element of the latter set. Also: $\{1,2\}\subset \{1,2,3\}$ , but $\{1,2\}\notin \{1,2,3\}$ .

The symbols $\in$ and $\subset$ are closely related because $x\in X\text{ if and only if }\{x\}\subset X\text.$

For all sets $A$ , $B$ , and $C$ :

$\emptyset\subset A$ .
$A\subset A$ .
$A=B\text{ if and only if } A\subset B\text{ and }B\subset A$ .
If $A\subset B$ and $B\subset C$ , then $A\subset C$ .

The proof follows almost immediately from the definition of the concepts of subset and empty set.

Power set

Power set

The set of all subsets of a set $X$ is called the power set of $X$ . We denote this as ${\Large\wp}(X)$ .

According to our naive set theory, ${\Large\wp}(X)$ is a set and for all $x$ , it holds that $x\in {\Large\wp}(X)$ if and only if $x\subset X$ . In the form of a formula: ${\Large\wp}(X)=\{Y\mid Y\subset X\}$

Examples

${\Large\wp}(\emptyset)=\{\emptyset\}$ ${\Large\wp}\bigl(\{a\}\bigr)=\bigl\{\emptyset, \{a\}\bigr\}$ ${\Large\wp}\bigl(\{a,b\}\bigr)=\bigl\{\emptyset, \{a\}, \{b\}, \{a,b\}\bigr\}$ ${\Large\wp}\bigl(\{a,b,c\}\bigr)=\bigl\{\emptyset, \{a\}, \{b\}, \{c\}, \\ \phantom{xxx}\{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\bigr\}$

If $X$ is a finite set with $n$ elements, then ${\Large\wp}(X)$ is a finite set with $2^n$ elements.

For all sets $A$ and $B$ :

If $A\subset B$ , then ${\Large\wp}(A)\subset {\Large\wp}(B)$ .
If ${\Large\wp}(A)\subset {\Large\wp}(B)$ , then $A\subset B$ .
If ${\Large\wp}(A)= {\Large\wp}(B)$ , then $A=B$ .

The proof follows almost immediately from the definition of the concepts subset and power set.