138 452 chapter 4 jointly distributed random

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Unformatted text preview: ite intervals of the real line. Let |A| denote the sum of the lengths of all intervals making up A, and |B | denote the sum of the lengths of all intervals making up B. The product set of A and B , denoted by A × B, is defined by A × B = {(u, v ) : u ∈ A, v ∈ B }, as illustrated in Figure 4.12. The total area of the product set, |A × B |, is equal to |A| × |B |. v [ [ B [ [ u A Figure 4.12: The product set A × B for sets A and B. The following proposition helps identify when a set is a product set. Proposition 4.4.3 Let S ∈ R2 . Then S is a product set if and only if the following is true: If (a, b) ∈ S and (c, d) ∈ S then (a, d) ∈ S and (c, b) ∈ S. (i.e. if (a, b) and (c, d) are points in S then so are (a, d) and (b, c).) Proof. (if) Suppose S is a product set, or in other words, suppose S = A × B for some sets A and B. Suppose also that (a, b) ∈ S and (c, d) ∈ S. Then a, c ∈ A and b, d ∈ B so (a, d), (c, b) ∈ A × B = S. (only if) Suppose S has the property that (a, b) ∈ S and (c, d) ∈ S imply that (a, d) ∈ S and (c, b) ∈ S. Let A...
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