Isye 2027

# 138 452 chapter 4 jointly distributed random

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

Ask a homework question - tutors are online