## Abstract

A set, *S*, will be defined as a “collection” (or “family”) of “objects” called *elements*. The statement “*s* is an element of *S*” will be denoted *s* ∈ *S*, and its negation will be denoted *s* ∉ *S*. The set with no elements will be called the *empty set*, and denoted ∅. Given a pair of sets, *S* and *T*, we say that *S* is a *subset* of T, and write *S* ⊂ *T*, if each element of *S* is an element of *T*. Again the negation of the statement will be denoted *S* ⊄ *T*. One obviously has ∅ ⊂ *S* for any set *S*. We write *S* = *T* if both *S* ⊂ *T* and *T* ⊂ *S*. *S* is called a *proper subset* of *T* if *S* ⊂ *T*, but *S* ≠ *T*. In this case one also says that the inclusion *S* ⊂ *T* is a *proper inclusion*. We shall constantly use the notation *S* = {*t* ∈ *T* : *P*(*t*)} to denote the set of all elements in *T* for which the property *P* holds. In most problems, all the sets we consider are subsets of a fixed (large) set, called the *universal set* or the *universe of discourse*, which we denote by *U*. We will usually assume that such a universe has been chosen, especially when complements of sets (to be defined below) are involved in the discussion. Before defining the basic operations on sets, let us introduce a notation which will be used throughout the book.

## Keywords

Equivalence Relation Commutative Ring Nonempty Subset Identity Element Choice Function## Preview

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