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Unformatted text preview: Assignment #8 ******************** Section 3.3********************* 7b) Let P = {A_a : a is a real number} (i) For each A_a, (0,a) is an element of A_a; thus A_a is not empty. Therefore, empty set is not in P. (ii) Let A_a and A_b be in P. Suppose A_a \intersect A_b is not empty. Then, there is an element (x,y) which is in both A_a and A_b, i.e., y = a  x^2 and y = b  x^2, ==> a = y + x^2 and b = y + x^2 ==> a = b ==> A_a = A_b. Therefore, if A_a and A_b are in P, then either A_a \intersect A_b is empty, or A_a = A_b. (iii) Let (x,y) be in RxR. Define k = y + x^2. Then k is in R, and y = k  x^2. Therefore, (x,y) is in A_k. Hence (x,y) is in the union of elements in P. Thus, RxR is a subset of the union of elements in P. Let (x,y) be in the union of elements in P. Then, (x,y) is in A_c or some real number c. By the definition of A_c, (x,y) is in RxR. Thus, the union of elements of P is a subset of RxR. Therefore, the union of elements of P is equal to RxR. From (i), (ii) and (iii), P is a partition of RxR. 7c) The point (s,t) in the plane R^2 is related to another point (u,v) iff both points are on the same vertical shift of the parabola y=x^2. 8b) Ordered pairs: { (1,1),(2,2),(3,3),(4,4),(5,5),(3,4),(4,3) } 9) The 2 partition elements are D_1={1,4,7} and D_2={2,3,5,6}....
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This note was uploaded on 03/18/2012 for the course MAT 108 taught by Professor Staff during the Winter '10 term at UC Davis.
 Winter '10
 Staff

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