Beginning Rigorous Mathematics/Sets and Functions

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We will now discuss some important set operations. Recall from the preliminaries that we only intuitively define a set to be a collection of distinct mathematical objects. There are much better and rigorous definitions of what sets are and form a subject for study all by itself, into which we will not delve.

Recall the meaning of the symbol "", which reads "is an element of" so that if is a set and an object, then a is a statement (which is either true of false, depending on whether is an element of ).


Sets and Set operations[edit]

In the following discussion, and denote any sets.

Definitions[edit]

The empty set: [edit]

We will assume that the empty set exists, and is denoted by . As the name suggests, the empty set contains no elements so that for any object the statement is false.

Set equality[edit]

Usually there is no ambiguity when we use the symbol "=" to refer to equality between sets. It is important that equality between sets is completely different to equality between numbers.

We define the logical statement "" to be true by definition when the statement "" (which reads " is contained in if and only if is contained in ") is true, and false otherwise. Intuitively this means that sets are equal if and only if they contain exactly the same elements. For example, is false since "" is false. It might be helpful to check the truth table to see that "" is a false statement. It should then be clear that "" is true.

Subsets[edit]

If every element of the set is an element of , we then say that is a subset of .

Rigorously, we say that the statement "" is true by definition when the statement "" (which reads "If is contained in then is contained in ") is true.

We have seen previously that the statement is false, however the statement is true. It should be clear that is false, since the statement "" is false.

Intersection[edit]

We define the intersection of sets by the symbol "". Rigorously we write "" which reads "The intersection of the sets and is by definition equal to the set which contains exactly the elements which are contained in both and ".

For example "".

We say that and are disjoint when .

Union[edit]

We define the union of sets by the symbol "". Rigorously we write "" which reads "The union of the sets and is by definition equal to the set which contains exactly the elements which are contained in either one of and ".

For example "".

Complement[edit]

To define the complement of the set we assume that the set is a subset of some universal set . We say " lives in ". Often the universal set is implicitly clear, for example when we are studying real analysis we often just assume or when studying complex analysis we assume .

We define the complement of a set by the superscript "". Rigorously "" which reads "the complement of in is the set of all elements which are contained in and not in ".

For example, if we assume then .

Relative complement[edit]

We define the relative complement of sets by the symbol "". Rigorously, "" which reads "the relative complement of in is by definition equal to the set containing all the elements contained in and is not contained in ".

For example "".

Some basic results[edit]

Lemma 1[edit]

. Which reads "A equals B if and only if A is a subset of B AND B is a subset of A"

proof As explained in the previous chapter, will be true by adjunction when both and are true.

We prove first .

Let , then by definition we have , which is logically equivalent to . By simplification we have that is true, and that is true. Therefore by definition , and are both true. By adjunction is true. Therefore is true.

Conversely, we prove .

Let . Then, by simplification, both and are true. By definition both and are true. By adjunction, is true, which is logically equivalent to . Then by definition is true. Therefore is true. QED.

Lemma 2[edit]

Lemma 3[edit]

and