Multivariable Advanced Calculus

12 SOME FUNDAMENTAL CONCEPTSthere is no way to determine to everyone’s satisfaction whether a given judge is anactivist. Also, just because something is grammatically correct does not meanit makes any sense. For example consider the following nonsense.S = {x ∈ set of dogs : it is colder in the mountains than in the winter} .So what is a condition?We will leave these sorts of considerations and assume our conditions make sense.The axiom of unions states that for any collection of sets, there is a set consisting of allthe elements in each of the sets in the collection. Of course this is also open to furtherconsideration. What is a collection? Maybe it would be better to say “set of sets” or,given a set whose elements are sets there exists a set whose elements consist of exactlythose things which are elements of at least one of these sets. If S is such a set whoseelements are sets,∪ {A : A ∈ S} or ∪ Ssignify this union.Something is in the Cartesian product of a set or “family” of sets if it consists ofa single thing taken from each set in the family. Thus (1, 2, 3) ∈ {1, 4, .2} × {1, 2, 7} ×{4, 3, 7, 9} because it consists of exactly one element from each of the sets which areseparated by ×. Also, this is the notation for the Cartesian product of finitely manysets. If S is a set whose elements are sets,∏AA∈Ssignifies the Cartesian product.The Cartesian product is the set of choice functions, a choice function being a functionwhich selects exactly one element of each set of S. You may think the axiom ofchoice, stating that the Cartesian product of a nonempty family of nonempty sets isnonempty, is innocuous but there was a time when many mathematicians were readyto throw it out because it implies things which are very hard to believe, things whichnever happen without the axiom of choice.A is a subset of B, written A ⊆ B, if every element of A is also an element of B.This can also be written as B ⊇ A. A is a proper subset of B, written A ⊂ B or B ⊃ Aif A is a subset of B but A is not equal to B, A ≠ B. A ∩ B denotes the intersection ofthe two sets A and B and it means the set of elements of A which are also elements ofB. The axiom of specification shows this is a set. The empty set is the set which hasno elements in it, denoted as ∅. A ∪ B denotes the union of the two sets A and B andit means the set of all elements which are in either of the sets. It is a set because of theaxiom of unions.The complement of a set, (the set of things which are not in the given set ) must betaken with respect to a given set called the universal set which is a set which containsthe one whose complement is being taken. Thus, the complement of A, denoted as A C( or more precisely as X \ A) is a set obtained from using the axiom of specification towriteA C ≡ {x ∈ X : x /∈ A}The symbol /∈ means: “is not an element of”. Note the axiom of specification takesplace relative to a given set. Without this universal set it makes no sense to use theaxiom of specification to obtain the complement.Words such as “all” or “there exists” are called quantifiers and they must be understoodrelative to some given set. For example, the set of all integers larger than 3. Orthere exists an integer larger than 7. Such statements have to do with a given set, inthis case the integers. Failure to have a reference set when quantifiers are used turns