2. Prove that the following relations are equivalence relations:(i) On R, withx~yifx-yEZ(ii) On Zx (Z - {0}), with (a, b)~ (c,d) if ad = bc.(We will need this later in the course.)

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Answered: On R, with r~yifx-yeZ ) On Zx (Z -…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 12E: Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and...

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Does 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?
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.
b) Find Af(x) if f(x) = (3x+1)(3x+ 4)(3x+7)... (3x+ 19). / c) Let A = {1,2,3}, B = {a, b, c}, and C = {x, y, z}. Consider the following relations R and S from A to B and from B to C, respectively. R = {(1,b), (2,a), (2, c)} and S = {(a, y), (b,x), (c, y), (c, z) }. %3D %3D Find the matrix representation of S°R
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11. Prove that the relation on L(V, V) given by similarity is an equivalence relation.
3. (a) Consider the set A = {x² E QxER). Determine whether A is closed under addition and multiplication. (b) Consider the relation R = {(x, y) = Z² : 21x² + y²}. Verify the properties of R.
3. For universe S = R² define relation ~ as: (a, b) ~ (c,d) ⇒ 2a-b2c - d. Prove that is an equivalence relation on R², or find a counterexample.
4. Let Let A= {2,4} and B={1,3,5} and define relations U, v, and W from A to B as follows. For all (x, y) e AXB, (x, y) EU means that y-x> 2 (x, y) eV means that アー1= W ={(2,5),(4,1),(2,3)} a. Draw arrow diagrams for U, V, and W b. Indicate whether any of the relations U, V, and W are functions. Explain your answers
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.
A// Which of these relations on {0, 1, 2, 3 } are equivalence relations, why? a) R1= {(0, 0). (1,1 ). (2, 2). (3 , 3)} b) R2= {(0, 0). (0, 2). (2. 0) . (2. 2). (2, 3). (3 , 2). (3 , 3)} c) R3= {(0, 0). (1 , 1). (1, 2). (2, 1 ). (2, 2). (3 , 3)} d) R4= {(0, 0). ( 1 ,1). (1 , 3). (2, 2). (2, 3). (3 , 1 ). (3 , 2).(3 , 3)} e) R5={(0, 0), (0, 1), (0, 2), ( 1 , 0) , (1,1), (1, 2), (2, 0). (2, 2), (3 , 3)} B// Find by relying on the above relationships 1- R3 o R2 2- R4 o R5 3- R1 UR3 4- R2OR5 5- R1 - R5 C// let A= {1,2}, B={2,4,6}, R: A? =B , then find R
8. Let A = (2, 4) and B = (1,3, 5) and define relations U, V, and W from A to B as follows: For all (x, y)e Ax B, (x, y) e U means that y-x> 2. (x, y)e V means that y-1=. W = (2, 5), (4, 1),(2, 3)}. a. Draw arrow diagrams for U, V, and W. b. Indicate whether any of the relations U, V, and W are functions.

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On R, with r~yifx-yeZ ) On Zx (Z - {0}), with (a, b)~ (c,d) if ad = bc. (We will need this later in the course.)