Set membership

Membership relates an object to a set: \(x \in A\) is read as “x belongs to A” and means that x is one of the elements of A. Its negation is \(x \notin A\). A set is determined only by its elements, so listing an object twice changes nothing: \(\{1,1,2\} = \{1,2\}\).

Worked example: let \(A = \{2, \{2\}, \varnothing\}\). Then \(2 \in A\), \(\{2\} \in A\) and \(\varnothing \in A\), but \(\{\varnothing\} \notin A\), because \(\{\varnothing\}\) is a set different from the three elements of A. In each case, check which objects are the elements of the set, not the objects nested inside them.

Typical mistake: writing \(2 \subseteq A\), which is meaningless because 2 is not a set. Membership goes from objects to sets, while inclusion \(\subseteq\) goes from sets to sets. Membership is primitive: it is not defined from other notions but postulated in the axioms.