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.