Introduction to Mathematical Analysis I - Second Edition

11 1.1.3 Prove the remaining items in Theorem 1.1.3 . 1.1.4 Let A , B , C , and D be sets. Prove the following. (a) ( A ∩ B ) × C = ( A × C ) ∩ ( B × C ) . (b) ( A ∪ B ) × C = ( A × C ) ∪ ( B × C ) . (c) ( A × B ) ∩ ( C × D ) = ( A ∩ C ) × ( B ∩ D ) . 1.1.5 Let A ⊂ X and B ⊂ Y . Determine if the following equalities are true and justify your answer: (a) ( X × Y ) \ ( A × B ) = ( X \ A ) × ( Y \ B ) . (b) ( X × Y ) \ ( A × B ) = [( X \ A ) × Y ] ∪ [ X × ( Y \ B )] . 1.2 FUNCTIONS Definition 1.2.1 Let X and Y be sets. A function from X into Y is a subset f ⊂ X × Y with the following properties (a) For all x ∈ X there is y ∈ Y such that ( x , y ) ∈ f . (b) If ( x , y ) ∈ f and ( x , z ) ∈ f , then y = z . The set X is called the domain of f , the set Y is called the codomain of f , and we write f : X → Y . The range of f is the subset of Y defined by { y ∈ Y : there is x ∈ X such that ( x , y ) ∈ f } . It follows from the definition that, for each x ∈ X , there is exactly one element y ∈ Y such that ( x , y ) ∈ f . We will write y = f ( x ) . If x ∈ X , the element f ( x ) is called the value of f at x or the image of x under f . Note that, in this definition, a function is a collection of ordered pairs and, thus, corresponds to the geometric interpretation of the graph of a function given in calculus. In fact, we will refer indistinctly to the function f or to the graph of f . Both refer to the set { ( x , f ( x )) : x ∈ X } . Let f : X → Y and g : X → Y be two functions. Then the two functions are equal if they are equal as subsets of X × Y . It is easy to see that f equals g if and only if f ( x ) = g ( x ) for all x ∈ X . It follows from the definition that two equal functions must have the same domain. Let f : X → Y be a function and let A be a subset of X . The restriction of f on A , denoted by f | A , is a new function from A into Y given by f | A ( a ) = f ( a ) for all a ∈ A . Definition 1.2.2 A function f : X → Y is called surjective (or is said to map X onto Y ) if for every element y ∈ Y , there exists an element x ∈ X such that f ( x ) = y . The function f is called injective (or one-to-one ) if for each pair of distinct elements of X , their images under f are also distinct. Thus, f is one-to-one if and only if for all x and x 0 in X , the following implication holds: [ f ( x ) = f ( x 0 )] ⇒ [ x = x 0 ] . If f is both surjective and injective, it is called bijective or a one-to-one correspondence . In this case, for any y ∈ Y , there exists a unique element x ∈ X such that f ( x ) = y . This element x is then denoted by f − 1 ( y ) . In this way, we already built a function from Y to X called the inverse of f .