(See exercise 24.)
Show that the relation
Describe the equivalence class
For each
(For
Prove that
Prove that
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Elements Of Modern Algebra
- 5. For each of the following mappings, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. (Compare these results with the corresponding parts of Exercise 4.) a. b. c. d. e. f.arrow_forwardDefine a relation on Z by x ~ y if and only if x + 2y is divisible by 3. Show that this relation is an equivalence relation.arrow_forwardFor a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence relation on R and describe the equivalence class containing 0.arrow_forward
- Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove that T is an equivalence relation on R. (b) Find the distinct equivalence classes of T.arrow_forwardConsider the following relation on R. For every x, y ∈ R, x ∼ y if andvonly if x −y ∈Q.(a) Prove that ∼ is an equivalence relation on R(b) What are the equivalent classes [0] and [√2]?arrow_forwardShow that the relation R on Z defined by a R b if and only if 5a − 3b is even, for a, b ∈ Z, is an equivalence relation.arrow_forward
- The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?arrow_forwardA relation R is defined on Z by xRy if and only if 8 divides 3x + 5y. Prove that R is an equivalence relation. Determine the distinct equivalence classes of R and their elements.arrow_forwardLet f : X → Y be a function. Define a relation R in X given byR = {(a, b): f(a) = f(b)}. Examine whether R is an equivalence relation or not.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,