Introduction to Mathematical Analysis I - 3rd Edition

11 Exercises 1.1.1 Prove the remaining items in Theorem 1.1.2. 1.1.2 ▶Let Y andZ 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 Dbe 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. Afunction 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 domainof f , the set Y is called the codomainof 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 eachx ∈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 gif and only if f (x) =g(x) for all x ∈X. Note that it is implicit in the definition that two equal functions must have the same domain and codomain. Let f : X →Y be a function and let Abe a subset of X. The restriction of f on A, denoted by f|A, is a new function fromAintoY given by f|A(a) =f (a) for all a∈A.

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