Last updates: 8 June 2011
A set is a collection of objects which are called elements. Write if is an element of a set .
The empty set is the set with no elements.
A subset of a set is a set such that if then . Write if is a subset of .
HW: Show that the emptyset is a subset of every set.
Two sets and are equal if and . Write if the set is equal to the set .
Let and be sets. The union of and is the set of all such that or ,
Let and be sets. The intersection of and is the set of all such that and ,
Let and be sets. The sets and are disjoint if .
Let and be sets. The sets The set is a proper subset of if and . Write if is a proper subset of .
The product of sets and is the set of all ordered pairs where and , More generally, given sets , the product is the set of all tuples such that .
The elements of a set are indexed by the elements of a set if each element of is labeled by a unique element of . In this case for will often be written to denote elements of so that . (CHECK THIS WORDING WITH THE ORIGINAL.)
Example. Let , , , and be the sets , , and . Then
Let and be sets. A function or map is given by associating to each element a unique element . Write Often in mathematics one will try to define a function without being explicitly sure if what has been defined is really a function. In order to check tha ta function is well defined one must check that
Let and be sets. Two functions and are equal if they satisfy
|if then .|
Let and be sets and let be a functions. Let . The restriction of to is the function defined by
A function is injective if satisfies the condition
|If and then .|
A function is surjective if satisfies the condition
|If then there exists such that .|
A function is bijective if it is both injective and surjective.
Let and be sets. The sets and are isomorphic, or have the same cardinality, if there is a bijective function from to .
Notation: Let be a set. Then
Note that even in the case where and it may be that .
Let be a set.
HW: Show that , that , and that .
Let and be functions. The composition of and is the function given by
Let be a set. The identity map on is the function given by
Let be a function. The inverse function to is a function such that
Let be a function. An inverse function to exists if and only if is bijective.
Examples. It is useful to visualise a function as a graph with edges connecting elements and . With this idea in mind we have the following examples.
(a) a function
(b) not a function
(c) not a function
(d) an injective function
(e) a surjective function
(f) a bijective function
Representing functions as graphs, the identity function looks like
(a) the identity function
In the pictures below, if the left graph is a pictorial representation of a function then the inverse function to , , is represented by the graph on the right; the graph for is the mirror-image of the graph for .
(b) the function
(c) the function
Graph (d) below, represents a function which is not bijective (it is neither injective nor surjective). The inverse function to does not exist in this case; graph (e) of a possible candidate, is not the graph of a function.
(d) the function
(e) not a function
Almost everything in mathematics is built from sets and functions. Groups, rings, fields, vector spaces, ... are all sets with specific functions which have special properties. In the society of mathematics, sets and functions are the individuals and the fascination is the way that the individuals, each one different from the others, interact.
The set of subsets of a set forms a partially ordered set under inclusion . The union and intersection make the set of subsets of into a Boolean algebra. Functions are the morphisms in the category 𝒮et of sets and products are products in the category 𝒮et.