Define a relation on the plane by setting
If either
Want to see the full answer?
Check out a sample textbook solutionChapter 1 Solutions
Topology
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardLet T be the set of all triangles in a plane with R a relation in T given byR = {(T1, T2) : T1 is congruent to T2}. Show that R is an equivalence relation.arrow_forwardThe relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and (x2, y2) in R^2a) Prove that R is an equivalence relation. Be sure to use the correctformat, including labelling b) ] Find the equivalence class [(2, 3)] and sketch it in the x-y plane.arrow_forward
- Let D = {(x, y) : x2 + y2 ≤ 1} be the unit disk in the plane. Place an equivalence relation ∼ on D by (x1, y1) ∼ (x2, y2) IFF x12 + y12 = x22 + y22. a) Find a continuous function. f:D → R (where R is the real number line) such that f((x1, y1)) = f((x2, y2)) IFF (x1, y1) ∼ (x2, y2) b) What space is D/~ homeomorphic to and why?arrow_forwardDefine a relation C from R to R as follows: For any (x,y) R x R, (x,y)C meaning that x2 + y2 = 1 Is (0, 0) C Is (1, 0) C? Is -2 C 0? Is 0 C (-1)?arrow_forwardShow that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are homotopy equivalent. (E^2 represents R^2 equipped with euclidean topology)arrow_forward
- Determine whether the relation R on the set of all realnumbers is reflexive, symmetric, antisymmetric, and/ortransitive, where (x, y) ∈ R if and only ifa) x + y = 0. b) x = ±y. c) x − y is a rational number. d) x = 2y. e) xy ≥ 0. f ) xy = 0. g) x = 1. h) x = 1 or y = 1.arrow_forwardLet R be the relation on R defined by xRy if there exists some n∈Zsuch that y=x⋅cny=x⋅cn. Let c = 6 and d be = 9arrow_forwardConsider the relation S on N defined by uSy⟺5|(y+u). Is it reflexive?arrow_forward
- Consider the relation R on Z with rule (a, b) in R iff a + 2b is even. Is R reflexive, symmetric, transitive?arrow_forwardConsider the mapping w = e^−iz of the domainD = { z belongs to C : Im(z) > 0 , −π/2 < Re(z) < 0 }Determine the image of D in the w-plane.arrow_forwardDefine a relation S E R × R by S = { (x,y) E R × R Ix-yE7}. prove that s is an equivalence relation of R.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning