## A.6 Sets Informally, a ***set*** is a collection of objects. We denote sets by capital letters such as *S*, and, if we enumerate all the objects in a set, we enclose them in braces. For example,  is the set containing the first four positive integers. The order in which we list the objects is irrelevant. This means that  are the same set—namely, the set of the first four positive integers. Another example of a set is  This is the set of the names of the days in the week. When a set is infinite, we can represent the set using a description of the objects in the set. For example, if we want to represent the set of positive integers that are integral multiples of 3, we can write  Alternatively, we can show some items and a general item, as follows:  The objects in a set are called ***elements*** or ***members*** of the set. If *x* is an element of the set *S*, we write *x* ∊ *S*. If *x* is not an element of *S*, we write *x* ∉ *S*. For example,  We say that the sets *S* and *T* are equal if they have the same elements, and we write *S* = *T*. If they are not equal, we write *S* ≠ *T*. For example,  If *S* and *T* are two sets such that every element in *S* is also in *T*, we say that *S* is a ***subset*** of *T*, and we write *S* ⊆ *T*. For example,  Every set is a subset of itself. That is, for any set *S*, *S* ⊆ *S*. If *S* is a subset of *T* that is not equal to *T*, we say that *S* is a ***proper subset*** of *T*, and we write *S* ⊂ *T*. For example,  For two sets *S* and *T*, the ***intersection*** of *S* and *T* is defined as the set of all elements that are in both *S* and *T*. We write *S* ∩ *T*. For example,  For two sets *S* and *T*, the ***union*** of *S* and *T* is defined as the set of all elements that are in either *S* or *T*. We write *S* ∪ *T*. For example,  For two sets *S* and *T*, the ***difference*** between *S* and *T* is defined as the set of all elements that are in *S* but not in *T*. We write *S* − *T*. For example,  The ***empty set*** is defined as the set containing no elements. We denote the empty set by Ø.  The ***universal set*** *U* is defined as the set consisting of all elements under consideration. This means that if *S* is any set we are considering, then *S* ⊆ *U*. For example, if we are considering sets of positive integers, then