Exercises 19-20 refer to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from A to B,
Let
State explicitly which ordered pairs are in
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Discrete Mathematics With Applications
- Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andBarrow_forwardLet and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.arrow_forwardLet A={a,b,c,d,e} and S, T, U and V relations on A where S = {(a,a), (a,b), (b,c), (b,d), (c,e), (e,d), (c,a)} T = {(a,b), (b,a), (b,c), (b,d), (e,e), (d,e), (c,b)} U= {(a,b), (a,a), (b,c), (b,b), (e,e), (b,a), (c,b), (c,c), (d,d), (a,c), (c,a)} V= {(a,b), (b,c), (b,b), (e,e), (b,a), (c,b), (d,d), (a,c), (c,a)} a) Which of the relations are reflexive? Justify your answer. b) Which of the relations are antisymmetric? Justify your answer. c) Find U V. d) Find T─S.arrow_forward
- Below are four coordinate axes for 3-space, labeled I, II, III, and IV. Which sets of axes adhere to the Right-Hand Rule? エ. I. Only I Only II Only II Only IV OI, II, and IIl; not IV O1, II, and IV; not III Both I and IV; not II or III Both III and IV; not I or II All of them None of themarrow_forwardLet A={a.b.c.d.e) and S, T, U and V relations on A where S {(aa), (ab), (b.c), (bd), (ae), (ed), (aa)} T= {(ab), (ka), (bc), (kd), lee), (de), (abl} U= {(a.b), (aa), (bc), (k.b), (ee), (ba), (ab), (Cuc), (d.d), (ac), (aa)} V= {(ab), (kc), beb), (ee), ka), (sub), (dd), (as), (aa} a) Which of the relations are reflexive? Justify your answer. b) Which of the relations are antisymmetric? Justify your answer. c) Find Un V. d) Find T-S.arrow_forwardLet A = (0,1,2,3). The relations R, S and T in the set A are defined as follows…R={(0,0),(0,1),(0,3),(1,0),(1,1),(1,3),(2,2),(3,0),(3,1),(3,3)}S ={(0,0),(0,2),(0,3),(2,3)}T={(0,1),(2,3)}Find the equivalent relation is:arrow_forward
- Let B = {0, 1, 2, 3} and the relations R, S, and T on Z are as follows: R = {(0, 1), (1,1), (2, 3),(3,3)} S= {(0, 0), (0, 2), (0, 3), (2, 3)}, T = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (1, 3),(2, 2), (3, 0), (3, 3)}, %3D T has . properties Seç. Seç. The equivalance relation is: R S has . properties transitive, symmetric but not reflexive reflexive, symmetric but not transitive R has . properties reflexive, symmetric and transitive reflexive, but neither symmetric nor transitive reflexive, transitive but not symmetric transitive, but neither symmetric nor reflexive symmetric, but neither reflexive nor transitivearrow_forwardLet R = { ( a, b ), ( a, d ), ( b, d ), ( c, a ), ( c, c ), ( d, b ), ( d, c ) } be a relation on the set { a, b, c, d }. Find R2 (i.e., R ∘ R), R3, (i.e., R2 ∘ R), R4, and R5.arrow_forwarda. Draw arrow diagrams for U, V, and W.b. Indicate whether any of the relations U, V, and W arefunctionsarrow_forward
- Please answer this questionarrow_forwardExercises to work. 1. Determine whether each of the following relations on the set {1, 2, 3, 4} is reflexive, symmetric, antisymmetric or transitive. (a) mi (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4) } (b) i (1, 1), (2, 2), (2, 1), (1, 2), (3, 3), (4, 4) } (c) (d) k(2, 4), (4, 2) } L(1, 2), (2, 3), (3, 4) } (e). L(1, 1), (2, 2), (3, 3), (3, 4) } (f) { (1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4) }arrow_forwardDetermine whether each of the following sets are linearly independent / dependent: (i) {(4, -4, 8, 0), (2, 2, 4, 0), (6, 0, 0, 2), (6, 3, -3, 0)} .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,