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Let U be the set of vertices in Fig. 1. Let
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Chapter 5 Solutions
Finite Mathematics & Its Applications (12th Edition)
- Based upon the hypothesis of a theorem, do the drawings of different students have to be identical same names for vertices, etc.?arrow_forwardConsider noncoplanar points A, B, C, and D. Using three points at a time such as A, B, and C, how many planes are determined by these points?arrow_forwardIn Discrete Matharrow_forward
- The Cartesian product of finite and nonempty sets can be illustrated by the intersections of a crossing-line pattern, as shown to the right. {f, g. h} x {x, y} (a) Explain why the number of intersection points in the crossing-line pattern for AxB is a•b, where a = n(A) and b=n(B). (a) Each of the lines coming from set A intersects each of the lines coming from set Barrow_forwardLet A = {0, 1, 2} and let B = {-2, 0, 1, 2}. Let r Ry if x > y. Find all (r, y) ER and list R as a set.arrow_forward3. Let A = {1,2, 3} and B = {1,2, 3,4}. The relations R1 = {(1,1), (2,2), (3,3)} and R2 = {(1,1), (1,2), (1,3), (1,4)} can be combined using set operations. Find the following: a. R1 U R2 b. Rị n R2 c. R1 \ R2 d. R2 \ R1arrow_forward
- The symmetric difference of set A and B, denoted by A∆B, is the set containing elements in eitherA or B, but not in both A and B. Answer the followings:arrow_forwardSuppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {Pi + q, P2 + q, P3 + q} is also affinely independent.arrow_forward2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a, b) : a € A and be B}. This can be extended naturally to a finite number of sets A1, A2, … , Aµ: A1 x Az x -..x Ak = {(a1, a2, ...,ak) : a; € Aj for 1< j< k}. %3D This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by II x X, = {s :1¬UX; : f6) e X; for all i e I iel That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X; for any i € I. Let {G¡}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = IIies Gi- %3D a. Show that G is a group under multiplication given by fg(i) = f(i)g(i) for all e I. We call G the direct %3D product of the family {G;}ie1-arrow_forward
- 3. Let A = {-1, –3, – 5}, B = {a, B, 7}, C = {x, y, z}. Consider the relations R from A to B and S from B to C respectively. R = {(-1,8), (–3, a), (–3, 7)} and S = {(a, y), (B, x), (7, y), (y, 2)} Find the following relations (a) Find the composition Ro S. (b) Find the matrices MR, Ms and MRos of the respective relations R, S and Ro S. (c) Compare M Ros to the product MRMS.arrow_forward5. Let A = {1, 2, 3}. Let G be a digraph and V(G) = pow(A). E(G) = {(u, v)|u < v}, where uc v means u is a proper subset (or strict subset) of v. a) Draw the graph G. b) Let v be the vertex corresponding to the set {2}. Find all elements of G reachable from v. c) Find two different topological sorts of G.arrow_forward
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