Discrete Math
12. Consider the relation Con R given by xCy if and only if x - y < 10. (a). Determine if C is reflexive. (b). Determine if C is symmetric. (c). Determine if C is transitive.
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
6. Define a relation on the plane by setting (xo, yo) < (x1, yı) if either yo - xz <yı - xy, or yo - xz = yı - xị and xo < X1. Show that this is an order relation on the plane and describe it geometrically.
10. (10 points) Define a relation on Z by setting x R y if xy is even. a. Give a counterexample to show that is not reflexive. b. Give a counterexample to show that R is not transitive.
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z) : x2 + y2 = p2, z = h}. 1 Show that for any r > 0 and h ER, Sony F. dx = 0
R × R | x < y} . This means that R 10. Let R< = {(x, y) relation on R. is the "less than" 95 (a) What is the domain of the relation R<? (b) What is the range of the relation R<? (c) Is the relation R a function from R to R? Explai. Note: Remember that a relation is a set. Consequently, we can talk about one relation being a subset of another relation. Another thing to...
Copy of R - 25x<2, -2sys 2 Consider the functions f(x,y)= 3 - 3 xy - x - y and the region (Then during the investigation of Absolute Extrema, we have to investigate(chose all possible options for values at all four intersection points to Investigate for Internal critical points and (-1,-1) is an internal critical point lla for Internal critical points and (1,1) is an internal critical point.Illa for critical points on four boundary.IVO critical points on three boundary VO
Find the cardinality of the set {(r, y) E R? : x² + y? < 1}.
5. Let F(x, y, z) = (yz, xz, xy) and define 2 Crin = {(x,y,z) : x2 + y2 = r2, 2 = h} Show that for any r > 0 and h ER, le F. dx = 0 Crih
Problem 10(20). Let x and y be vectors in R". Prove that |x"y| < ||x|||y- No work, no credit, messy work, no credit, missed steps, no credit disorganized work, no credit.