11 Exercises 1.1.1 Prove the remaining items in Theorem 1.1.2. 1.1.2 ▶ Let Y and Z be subsets of X. Prove that (X \Y) ∩ Z = Z \ (Y ∩ Z). 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 Defnition 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: (i) For all x ∈ X there is y ∈ Y such that (x,y) ∈ f . (ii) 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 defned by {y ∈ Y : there is x ∈ X such that (x,y) ∈ f }. It follows from the defnition 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 valueof f atx or the image of x under f . Note that, in this defnition, 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. Note that it is implicit in the defnition that two equal functions must have the same domain and codomain. 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.
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