Define a relation from R to R by saying that (x,y) ES if and only if...
Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only if either x-a or y- b. Prove or disproof, isan equivalence relation? If so, write down all the equivalence classes.
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
Define a relation < on Z by m <n iff |m| < |n| or (\m| = |n| 1 m <n) (a) Prove that < is a partial order on Z. (b) A partial order R on a set S is called a total order (or linear order) iff (Vx, Y ES)(x + y + ((x, y) E R V (y,x) E R)) Prove that is a total order on Z. (c) List the following elements in <-increasing order. –5, 2,...
3) Define the relation <on R via x < y if and only if xy < 10. Show that is symmetric. (20 points)
Define a relation R on N x N by R = {(x,y) | x ε N, y ε N and x+y is even} Prove or disprove: R is an equivalence relation.
2) We define the relation R between two elements of S as: R= {(x,y) ifx is a subset of y or x-y) Show that R is a partial order.
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
4. Consider the relation on the positive integers xRy if and only if x x+y (a) List three ordered pairs from this relationship (b) Is R reflexive? Prove your answer (c) Is R symmetric? Prove your answer (d) Is R anti-symmetric? Prove your answer (e) Is R transitive? Prove your answer.
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...