### Sets: Introduction

### On the content of Sets

This chapter deals with one of the most fundamental concepts of mathematics: sets. A set is understood to be a collection of objects. The word object refers to anything from people to the universe, from a number to a word. As to be expected, we will be using it mostly for mathematical expressions. Sets are a way of distinguishing particular objects (those in the set) from other objects (those outside). The objects inside a set are called its elements.

We deal with two ways of identifying the elements of a set. The simplest way, which works well for sets of moderate size, is enumeration. It is common to use curly brackets to enclose the elements. Thus, #\{1,2,3,4,5\}# denotes the set of integers between #1# and #5#. For instance, the real numbers are impossible to enumerate, so enumeration is not always an option. Even when a set can be enumerated, an enumeration notation may not be the best option for identifying the elements of the set precisely. A typical example is the set of even integers which contains infinitely many elements. Enumeration by use of dots of this set, like #\{\ldots,-6,-4,-2,0,2,4,6,\ldots\}#, is not very exact and may leave room for misinterpretations.

Our second method defines a set #A# by means of a statement that decides, for each object, whether or not it is an element of #A#. The so-called set builder method describes this set precisely. For instance the set of even integers can be described as the set of all integers that are an integer multiple of #2#. Again, brackets are used in common notation for this set, for example \[\left\{\left.n \text{ element of }\mathbb{Z}

\,\right|\,n = 2\cdot m\text{ for some integer }m\right\}\]

A complication in defining sets is the fact that the statements used to define a set should lead to a decision procedure whether or not an object is an element of the set, or, at the very least, not to a contradiction. It will not be a surprise that the chapter

Operations on sets are the next topic. An example of such an operation is the intersection: when given two (or more) sets #A# and #B# (not necessarily different from one another), the intersection of #A# and #B# is the set #C# of all objects that are both an element of #A# and an element of #B#. To illustrate, consider the sets #A=\{1,2,3,4\}# and #B=\{2,4,6,8,10\}#, the intersection of #A# and #B# is # \{2,4\}#. Compositions of such operations obey some rules which are, of course, also discussed.

Another operation discussed is the construction of the set of all pairs of elements, one from a specified set #A# and the other from a specified set #B#. If both #A# and #B# are the set of real numbers, this construction coincides with the real plane. A set of elements (not necessarily all) from this so-called

The idea of a set is quite simple, but the mathematical ramifications are quite complex. This chapter is of an introductory nature; we only discuss the basics.

We deal with two ways of identifying the elements of a set. The simplest way, which works well for sets of moderate size, is enumeration. It is common to use curly brackets to enclose the elements. Thus, #\{1,2,3,4,5\}# denotes the set of integers between #1# and #5#. For instance, the real numbers are impossible to enumerate, so enumeration is not always an option. Even when a set can be enumerated, an enumeration notation may not be the best option for identifying the elements of the set precisely. A typical example is the set of even integers which contains infinitely many elements. Enumeration by use of dots of this set, like #\{\ldots,-6,-4,-2,0,2,4,6,\ldots\}#, is not very exact and may leave room for misinterpretations.

Our second method defines a set #A# by means of a statement that decides, for each object, whether or not it is an element of #A#. The so-called set builder method describes this set precisely. For instance the set of even integers can be described as the set of all integers that are an integer multiple of #2#. Again, brackets are used in common notation for this set, for example \[\left\{\left.n \text{ element of }\mathbb{Z}

\,\right|\,n = 2\cdot m\text{ for some integer }m\right\}\]

A complication in defining sets is the fact that the statements used to define a set should lead to a decision procedure whether or not an object is an element of the set, or, at the very least, not to a contradiction. It will not be a surprise that the chapter

*Logic*is of use here.Operations on sets are the next topic. An example of such an operation is the intersection: when given two (or more) sets #A# and #B# (not necessarily different from one another), the intersection of #A# and #B# is the set #C# of all objects that are both an element of #A# and an element of #B#. To illustrate, consider the sets #A=\{1,2,3,4\}# and #B=\{2,4,6,8,10\}#, the intersection of #A# and #B# is # \{2,4\}#. Compositions of such operations obey some rules which are, of course, also discussed.

Another operation discussed is the construction of the set of all pairs of elements, one from a specified set #A# and the other from a specified set #B#. If both #A# and #B# are the set of real numbers, this construction coincides with the real plane. A set of elements (not necessarily all) from this so-called

*Cartesian product*of #A# and #B# will be interpreted as a*relation*between the two sets. For sets of manageable size, we will show how such a relation can be visualized. Finally, another fundamental mathematical concept, that of a*function*, will be interpreted within the context of relations.The idea of a set is quite simple, but the mathematical ramifications are quite complex. This chapter is of an introductory nature; we only discuss the basics.

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