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 [As =(−1,1) and \ As = {0}. s∈J s∈J 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 B be a subset of X. Then the following hold: �T T (i) B∪ i ∈I Ai = i ∈I (B ∪ Ai). �S S (ii) B∩ i ∈I Ai = i ∈I (B ∩ Ai). �T S (iii) B\ i ∈I Ai = i ∈I (B\ Ai). �S T (iv) B\ i ∈I Ai = i ∈I (B\ Ai). �T c S = (v) i ∈I Ai i ∈I Ac i . �S c T c = . (vi) i ∈I Ai i ∈I Ai �T T Proof of (i): Let x ∈ B ∪ i ∈I Ai . Then x ∈ B or x ∈ i ∈I Ai. If x ∈ B, then x ∈ B∪ Ai for all i ∈ I T T and, thus, x ∈ i ∈I (B ∪ Ai). If x ∈ i ∈I Ai, then x ∈ Ai for all i ∈ I. Therefore, x ∈ B ∪ Ai for all i ∈ I T �T T and, hence, x ∈ i ∈I (B ∪ Ai). We have thus showed B ∪ i ∈I Ai ⊂ i ∈I (B ∪ Ai). T �T Now let x ∈ i ∈I (B ∪ Ai). Then x ∈ B∪ Ai for all i ∈ I. If x ∈ B, then x ∈ B∪ i ∈I Ai . If x ̸ ∈ B, T �T then we must have that x ∈ Ai for all i ∈ I. Therefore, x ∈ i ∈I Ai and, hence, x ∈ B∪ i ∈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 a and b, we will denote by (a,b) the ordered pair where a is the frst element and b is 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 A and B, the Cartesian product of A and B is the set defned by A× B := {(a,b) : a ∈ A and b ∈ B}. ■ Example 1.1.2 If A = {1,2} and B = {−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 defnition of ordered pair in terms of sets see [Lay13]
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