Define an equivalence relation on R by (x,y,z) ∼ (u,v,w) whenever x +y +z = u...
Prove that the following relation R is an equivalence relation on the set of ordered pairs of real numbers. Describe the equivalence classes of R. (x, y)R(w, z) y-x2 = z-w2
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.
1. Define a relation on Z by aRb provided a -b a. Prove that this relation is an equivalence relation. b. Describe the equivalence classes. 2. Define a relation on Z by akb provided ab is even. Use counterexamples to show that the reflexive and transitive properties are not satisfied 3. Explain why the relation R on the set S-23,4 defined by R - 11.1),(22),3,3),4.4),2,3),(32),(2.4),(4,2)) is not an equivalence relation.
Determine if {(x,y) | x divides 2-y} is an equivalence relation on {1,2,3,4,5}. List the equivalence classes Determine if {(x,y) | x and y are both even or x and y are both odd} is an equivalence relation on {1,2,3,4,5}. List the equivalence classes. Determine if {(x,y) | x and y are the same height} is an equivalence relation on all people Determine if {(x,y) | x and y have the same color hair} is an equivalence relation on all...
4.let U= {q,r,s,t,u,v,w,x,y,z}; A= {q,s,u,w,y};and C={v,w,x,y,z,}; list the members of the indicated set , using set braces A'u B A.{Q,R,S,T,V,X,Y,Z} B.{S,U,W} C.{R,S,T,U,V,W,X,Z} D.{Q,S,T,U,V,W,X,Y}
mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism. mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
Show the following is an equivalence relation: Define the relation ∼ on Z by a ∼ b iff a − b = 7k for some k ∈ Z. Then ∼ is an equivalence relation
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.