An Introduction to Sets Background The concept of a set is intuitive We view a set as a collection of objects Nothing more nothing less However when we begin to work with sets in a practical way we hope that we can find a relationship between those objects It is the relationship that we try to verbalize or otherwise quantify Let s look at some examples Here s a picture of a room To begin the process we could describe a set this way Let R be the set of all the objects pictured in this room The words objects items and elements are used interchangeably Notice that since we have the picture this is a well defined set Without this picture the description is very vague Also if I say that R is all the objects in this room I would be seriously incorrect There are many other items in this room not shown in the picture The word pictured is critical to this definition The Set R A well defined set must convey the same sense of the collection to all users To help us do that we list an element of a set exactly one time in its description We ll see why when we get around to counting processes However the order of the listing doesn t matter Now let s talk about membership in a set We regularly talk about the elements of a set Since we used R as the name for the set of objects pictured it is natural to let r represent an item pictured in the room So some of our choices for r could be a television a Japanese doll a model ship and a small spoiled dog named Princess Princess is a Pomeranian but she does not know she is a dog She invited herself into this picture If we compile these elements into a set by themselves we have a subset of the original set R Let s call it A Set Notation The following phrases and symbols are regularly used in talking about sets The phrase the set consisting of is represented by braces Example R all objects pictured in the room Example A television Japanese doll model ship small spoiled dog The phrase is an element of or more simply is in is represented by the E like symbol 0 Example Princess 0 R means Princess is one the objects pictured in the room Example television 0 R means A television is one the objects pictured in the room Arizona State University Department of Mathematics and Statistics 1 of 4 The phrase is a subset of is represented by the C like symbols d or f Think is Contained in The d symbol means that there is at least one element of the larger set not in the subset This is also called a proper subset A d R means The televison doll model ship and small spoiled dog are each members of the collection of objects pictured in the room It also demands that something else be in the picture that is not included in A The hanging lamp in the corner is sufficient to make this true The f symbol allows for the two sets in question to be equal This is also called an improper subset Example A f R allows for either of these statements The televison doll model ship and small spoiled dog are each members of the collection of objects pictured in the room but other objects may be pictured The sets are not equal or possibly The televison doll model ship and small spoiled dog are all of the members of the collection of objects pictured in the room The sets are equal Certainly both of the statements cannot be true at one time but as you learned when we studied logic when two or more statements are connected by the word or only one of the connected statements must be true for the entire sentence to be true Example Princess d R means The set with only Princess in it is a subset of R The concept of not is regularly conveyed by a slash across a symbol So and mean not an element of and not a subset of respectively You should recognize the slashed P symbol to the right as do not park here or else Example Let s build a set called B television model ship small spoiled dog cat Looking closely at the picture you can see there is no cat pictured Princess has declared no cat may ever come into that room and that is that Then cat R cat R and B R These statements tell us that there is no cat in the collection and the subset with only a cat in it and the set B which is described with a cat as a member are not subsets of R Universal Sets When speaking a set of objects it is often useful to define a universal set This is in a sense the smallest set having all the elements that interest us In the previous discussion the universal set could be R itself We didn t talk about anything else Suppose we had a series of photos of rooms from the same house We could then use universal the set We could also write a series This chain is true Princess 0 Princess d A f R f U Notice the use of the 0 an However remembering Princess s attitude about cats this chain is false B f R f U Since B has a ca it it cannot be a subset of R so the chain cannot be true Hence B R f U can be an appropriate representation of the relationship among B R and U 2 of 4 Arizona State University Department of Mathematics and Statistics Describing Sets We have a final bit of detail about describing sets We have a number of very good ways to describe them We could do much as we have before when we verbalized a description for R We might just list the elements of the set when it is small enough This is called a roster method We did this for both A and B above but not for R which has way too many elements in it The next method is slightly more technical It is called set builder notation Remember that somewhere along the line we said r could be any element in R Let s see how it s used in set builder notation The set R r r is pictured in the room is complete set builder notation It is read R is the set of elements r such that r is pictured in the room The is read such that or where This style of describing sets is particularly good when talking about the world of numbers Example Suppose we need to describe all the numeric values between zero and one The roster method is impossible There are infinitely many We could just say Let X represents all numbers between zero and one This is a perfectly good description but it just doesn t …