Introducing Mappings

    Mappings and functions are synonyms; that is, mappings assign every element in one set a unique value in another set. Mappings are one of the most fundamental concepts in all of mathematics and can be used not only to define groups in an abstract manner, but also to clarify the meaning of statements such as "two groups have essentially the same structure".

The set notations for the following commonly used sets are:

    (i) is the set of natural numbers  
    
    (ii) is the set of integers
    
    (iii) is the set of rational numbers  
    
    (iv) is the set of real numbers.
    
    (v) is the set of complex numbers.
    

Definition (Cartesian Products and Relations) Let and be sets. Then is the Cartesian product of and and subsets of are relations.

Definition (Mappings) A mapping is a set a set , and a subset of such that

    (i) if then there is an element such that
    
    (ii) if and then
    
In other words, is a mapping (or correspondence) from to if it satisfies conditions and (ii) (and is denoted by ); namely, assigns to every element of a unique element of The set is the domain, the set is the codomain, and the subset of is the graph of the mapping. Two mappings and are equal if they have the same domain, same codomain, and the same graph. In particular, a mapping from a set to itself with the property is the identity mapping on .  

Example (Types of Mappings) The following are not mappings:
    
    


and the following are mappings:
    

Definition (Types of Mappings) If then

    (i) if then is the image of under   
    
    (ii) if then the set is the pre-image of under
    
    (iii) if then is surjective (or onto);
    
    (iv) if for all then is injective  (one-to-one); and
    
    (v) if is both injective and surjective, then is bijective.
    
    

Example (Types of Mappings) The mapping defined by is one-to-one but not onto. The mapping defined by is onto but not one-to-one.

Proposition (Mappings of Sets) If and and are subsets of then

    (i)
    
    (ii) and
    
    (iii) is one-to-one if and only if
    
    Proof. (i): By definitions of image of a subset, and union of sets:
    
    
     such that
     or   such that
     such that or such that
    
            
    (ii): If then there exists such that So and and therefore, and By definition of intersection and subset,
    (iii): Suppose is one-to-one. By (ii), it suffices to show that If then there exists an such that and there exists an such that Since is one-to-one, and thus Whence, Conversely, assume for any subsets and of If in and and then is empty and thus so is So there is no element such that Therefore, and so is one-to-one.

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