Set Theory 2.1 Sets 2.1.1 Examples of Sets and Their Elements

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Abstract	The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define a set as a collection of objects that is well-defined. That is, it is possible to determine if an object is to be included in the set or not. Frequently a set is denoted by a capital letter, like S. Objects in the set collection are known as elements of S. If x is one of these elements, then we write that x ∈ S. Similarly, if an object x is not a part of the set, then we write x ∈ S. The most basic set is the collection of no objects. This set, known as the empty set or null set, is denoted by ∅ or {}, to indicate that it contains no elements. In Mathematics, the most frequently encountered sets are various collections of types of real numbers. The below such sets are of key importance. · The set of natural numbers, denoted by N, is the set of all non-negative whole numbers. Thus, we can list off a few of the elements by writing Note that some mathematicians have the natural numbers defined as the set of all positive whole numbers. · The set of integers, denoted by Z, is the set of all whole numbers. Thus, we can list them off as · The set of all real numbers is denoted by R. It is important to note that sets are unordered objects. Thus, the set {a, b, c} and the set {b, c, a} are considered the same because they contain the same elements, even if they are listed in different orders.
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