Introduction to Mathematical Analysis I - 3rd Edition

10 1.1 Basic Concepts of Set Theory (b) Here we let the index set be J = (0,1] and for each s ∈J we have As = (−s,s). Then [ s∈J As = (−1,1) and \ s∈J As ={0}. The proofs of the following properties are similar to those in Theorem 1.1.2. We include the proof of part (i) and leave the rest as an exercise. Theorem 1.1.3 Let {Ai : i ∈I}be an indexed family of subsets of a universal set X and let Bbe a subset of X. Then the following hold: (i) B∪ Ti∈I Ai =Ti∈I(B∪Ai). (ii) B∩ Si∈I Ai =Si∈I(B∩Ai). (iii) B\ Ti∈I Ai =Si∈I(B\Ai). (iv) B\ Si∈I Ai =Ti∈I(B\Ai). (v) Ti∈I Ai c =Si∈I Ac i . (vi) Si∈I Ai c =Ti∈I Ac i . Proof of (i): Let x ∈B∪ Ti∈I Ai . Thenx ∈Bor x ∈Ti∈I Ai. If x ∈B, thenx ∈B∪Ai for all i ∈I and, thus, x ∈Ti∈I(B∪Ai). If x ∈Ti∈I Ai, then x ∈Ai for all i ∈I. Therefore, x ∈B∪Ai for all i ∈I and, hence, x ∈Ti∈I(B∪Ai). We have thus showed B∪ Ti∈I Ai ⊂Ti∈I(B∪Ai). Now let x ∈Ti∈I(B∪Ai). Then x ∈B∪Ai for all i ∈I. If x ∈B, then x ∈B∪ Ti∈I Ai . If x̸ ∈B, then we must have that x∈Ai for all i ∈I. Therefore, x∈Ti∈I Ai and, hence, x∈B∪ Ti∈I Ai . This proves the other inclusion and, so, the equality. □ We want to consider pairs of objects in which the order matters. Given objects aand b, we will denote by (a,b) the ordered pair where ais the first element and bis the second element. The main characteristic of ordered pairs is that (a,b) = (c,d) if and only if a=c and b=d. Thus, the ordered pair (0,1) represents a different object than the pair (1,0) (while the set {0,1}is the same as the set {1,0})1. Given two sets AandB, the Cartesian product of AandBis the set defined by A×B:={(a,b) : a∈Aandb∈B}. ■ Example 1.1.2 If A={1,2}andB={−2,0,1}, then A×B={(1,−2),(1,0),(1,1),(2,−2),(2,0),(2,1)}. ■ Example 1.1.3 If A and B are the intervals [−1,2] and [0,7] respectively, then A×B is the rectangle [−1,2] ×[0,7] ={(x,y): −1≤x ≤2, 0≤y ≤7}. We will make use of cartesian products in the next section when we discuss functions. 1For a precise definition of ordered pair in terms of sets see [Lay13]

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