Consider the poset .Z Z; / where is the product order; that is, .x; y/ .x0 ; y0 / if and only if x x 0 and y y 0 . See Exercise 54.13. a. In this poset, calculate .1; 2/ ^ .4; 0/ and .1; 2/ _ .4; 0/. b. For arbitrary .x; y/ and .x0 ; y0 / in ZZ, give a formula for .x; y/ ^ .x0 ; y0 / and for .x; y/ _ .x0 ; y0 /. Verify that your formula is valid and conclude that this poset is a lattice. c. Show that this lattice satisfies the distributive properties (presented in the previous exercise).

Lecture 32 The Deﬁnite Integral (Section 5.2) Def. If f is deﬁned for a ≤ x ≤ b,d e[ a,b]no n subintervals of equal width ∆x = Let x 0= a),x ,1 ,2 .,x n(= b)behedisfe subintervals and let x be the right endpoint...