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- Let 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_forwardState whether the statement is true or false :'' All relations are mapping but the converse is not true ''.arrow_forwardDetermine if the relation R = {(1,3), (1,4), (2,3), (2,4), (3,4)} is reflexive, symmetric, antisymmetric, or transitive.arrow_forward
- Consider the plane (P) related to x-axis and y-axis system. Let R be the binary relation on (P) defined by: М (х, у) R M'(x', у') —x-у%3х' - у' |(a) Prove that R is an equivalence relation on the set of points of (P). (b) Interprete geometrically the equivalence class of the point A (1, 2). (c) Describe geometrically the set (P)/R.arrow_forwardLet A={1,2,3,4,5,6} and consider the following 3 subsets of A. A, = {1,3,5}; A2 = %3D {2,4, 6}; A3 = {3, 6}. Define the relation R on set A as follows: x R y if and only if { x, y} C A or { x, y}C A, or { x,y}C A. That is, x R y if and only if x and y are both in A, or x and y are both in A2 or x and y are both in A3.arrow_forwardSuppose there are two transitive relations R, S Disprove the claim that the composed relation So R must be transitive.arrow_forward
- For universe S = R², define relation ~= as: (a,b) ~= (c,d) < 2a - b = 2c - d Prove that ~= is an equivalence relation on R², or find a counterexample.arrow_forwardShow that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of thepoint Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.arrow_forwardconsider 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_forward
- The real line R is divided into subsets X1, X2, X3 where X1 = (-00, – 7], X2 = [-7,1), and X3 = [1, 00). Can X1, X2, X3 be equivalence classes with respect to some equivalence relation on R? O No, they can't be equivalence classes for some equivalence relation since X1n X2 + 0. O No, they can't be equivalence classes for some equivalence relation since any equivalence relation on infinite set has infinitely many different equivalence classes. O Yes, these sets can be equivalence classes for some equivalence relation since X1 U X2 U X3 = R. O Yes, these sets can be equivalence classes for some equivalence relation since X2n X3 = 0.arrow_forwardDefine a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈N}. Depict this relationship using roster form. Write down the domain and the range.arrow_forwardLet A = {1, 2, 3, 4, 5} and R be the relation defined by R = {(1,1), (2,2), (2,4), (2,5), (3,3),(4,2), (4,4)}. Justify whether relation R fulfill the property of: i) Reflexive. ii) Symmetric.(iii) Anti-symmetric.(iv) Transitive.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Elementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,Algebra for College StudentsAlgebraISBN:9781285195780Author:Jerome E. Kaufmann, Karen L. SchwittersPublisher:Cengage Learning