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_forwardDoes the relation is a brother of have a reflexive property consider one male? A symmetric property consider two males? A transitive property consider three males?arrow_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_forward
- consider the relation R={(1,a),(1,b), (3,b),(3,d), (4,b)} form X={1,2,3,4} and Y={a,b,c,d}. Thus G={y|1Ry} and H={y|2Ry} are equal toarrow_forwardLet R and S be symmetric relations. Show: R ◦ S symmetric ⇔ R ◦ S = S ◦ Rarrow_forwardLet A = {a, 6, 7, 8}. Determine whether relation R₂ = {(a, b), (a, a), (B,B),(a, y), (y, a),(1,7), (6, 6), (8, a), (B, 6)}is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, and/or transitive on Α Χ Α. Check the box below if the property satisfies the relation. Reflexive Irreflexive Symmetric Asymetric Antisymmetric Transitive The relation does not satisfy any of the given properties.arrow_forward
- This question refers 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 set A to a set B, RUS = {(x, y) EAXB | (x, y) ER or (x, y) ES) Rns={(x, y) EAxBI (x, y) ER and (x, y) E S). Let A = (-2, 2, 5, 7) and B (2, 5) and define relations R and S from A to B as follows: For every (x, y) EA x B, xRylxl = lyl and xSyx-y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RUS, and R n S. (Enter your answers as comma-separated lists of ordered pairs.) AxB= R = S = RUS = Rns=arrow_forwardLet C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a, b = R}. Define addition + and multiplication of such pairs by (u, v) + (x, y) = (u + x, v+y) and (u, v) • (x, y) = (ux — vy, uy + vx) R. Together they form a triple . for all u, v, r, y (a) Show that multiplication is associative in . (b) Show that every element (a, b) € C has a negative, and every element (a, b) € C# has an inverse. (c) Prove or disprove: The system of real numbers R is isomorphic to the system . Here, 0 R is the zero of R. (d) True or false? Justify your answer: The triple C, +, > must contain a subfield isomorphic to R.arrow_forwardThis question refers 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 set A to a set B, RUS= {(x, y) E A × B| (x, y) ERor (x, y) E S} RO S= {(x, y) E A x B | (x, y) ERand (x, y) E S}. Let A = {-4, 4, 7, 9} and B = {4, 7}, and define relations R and S from A to B as follows. For every (x, y) E Ax B, x R y x| = [y] and x Sy x - y is even. Which ordered pairs are in A x B, R, S, R U S, and RN S? (Use set-roster notation to enter your answers.) Ax B = R = S = RUS = ROS =arrow_forward
- . Let R1, R2, R3 be relations on { a, b, c, d, e }. Find the transitive closures, given their definitions below. i) R1 = { (a, c),(b, d),(c, a),(d, b),(e, d) } ii) R2 = { (b, c),(b, e),(c, e),(d, a),(e, b),(e, c) } iii) R3 = { (a, b),(a, c),(a, e),(b, a),(b, c),(c, a), (c, b),(d, a),(e, d) }arrow_forwardLet A = {1, 2, 3, 4], B = {a, b, c}, C = {x, y, z}. Consider the relations R from A to B and S from B to C as follows: R = {(1, b), (3, a), (3, b), (4, c)} and S = {(a, y), (c, x), (a, z)} (i) Draw the diagrams of R and S. (ii) Find the matrix of each relation R, S (composition) RoS. (iii) Write R −1 and the composition RoS as sets of ordered pairs. (b) Find x and y where: (i) (x + 2, 4) = (5, 2x + y); (ii) (y − 2, 2x + 1) = (x − 1, y + 2)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
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