For each of the following sets, determine whether 2 is an...

For each of the following sets, determine whether 2 is an element of that set. $$ \begin{array}{l}{\text { a) }\{x \in \mathbf{R} | x \text { is an integer greater than } 1\}} \\ {\text { b) }\{x \in \mathbf{R} | x \text { is the square of an integer }\}} \\ {\text { c) }\{2,\{2\}\}} \\ {\text { e) }\{\{2\},\{2,\{2\}\}\}} & {\text { f) }\{\{2\},\{\{2\}\}\}}\end{array} $$

For each of the following sets, determine whether 2 is an element of that set.
$$
\begin{array}{l}{\text { a) }\{x \in \mathbf{R} | x \text { is an integer greater than } 1\}} \\ {\text { b) }\{x \in \mathbf{R} | x \text { is the square of an integer }\}} \\ {\text { c) }\{2,\{2\}\}} \\ {\text { e) }\{\{2\},\{2,\{2\}\}\}} & {\text { f) }\{\{2\},\{\{2\}\}\}}\end{array}
$$

Key Concepts

Set Membership

Set membership is a fundamental concept in set theory that describes when an object is an element of a set. This is denoted by the symbol '?'. Determining set membership involves verifying whether a given object exactly matches an element in the set according to the set's definition and the rules of equality that apply in the number system or other contexts.

Set Builder Notation

Set builder notation is a concise way of specifying a set by stating a property or condition that its members must satisfy. Instead of listing all elements, it defines a set by a rule. This is particularly useful for sets with infinitely many elements or when the pattern of membership is more important than individual elements.

Nested Sets and the Distinction Between Elements and Subsets

Nested sets occur when elements of a set are themselves sets. In this context, it is crucial to distinguish between an element and a subset. An integer like 2 is not the same as the set {2}. This distinction is important because membership checks involve exact matches: 2 being an element of a set does not automatically mean that a set containing 2 is equivalent to the object 2.

Equality in Set Theory

Equality in set theory is determined by the principle that two sets are equal if and only if they have exactly the same elements. Conversely, when evaluating membership, care must be taken because objects may appear similar but are not identical. For example, the number 2 and the set containing 2, {2}, are entirely different even though there is a relation between them, which emphasizes the need for careful consideration of both elements and their structure within sets.