Question

In: Advanced Math

Question: Consider the relation R on A defined by aRb iff 1mod4 = bmod4 a)Construct the...

Question: Consider the relation R on A defined by aRb iff 1mod4 = bmod4

a)Construct the diagraph for this relation

b)show that R is an equivalence relation

Part B: Now consider the relation R on A defined by aRb iff a divides b (Divides relation)

c) Show that R is partial ordering

d) Contruct the hasse diagram for this relation

Solutions

Expert Solution

Part-A:

The relation is

(i)Reflexive

since

(ii):Symmetric

Suppose that

Hence

(iii):Transitive

Suppose that

Hence

Again

Suppose that

Hence

Thus we have

Hence

From (i),(ii),(iii) R is equivalence relation

If we consider the vertices as

then using above we find that

Prt-B:

(I) Reflexive since

(II) Anti-Symmetric since

(III)Transitive since

Hence equivalence

Hasse Diagram:

Consider the set with the above relation:

Colby Messinger answered 2 years ago
Next > < Previous

Related Solutions

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R...
Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R is an equivalence relation, that is, prove that it is reflexive, symmetric, and transitive. Determine the equivalence classes of this relation. What members are in the class [2]? How many members do the equivalence classes have? Do they all have the same number of members? How many equivalence classes are there? Question 2. Equivalence Relation 2 Consider the relation from last week defined as:...
Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show that...
Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show that R is an equivalence relation. Describe the elements of the equivalence class of 2/3.
Let R be the relation on the set of people given by aRb if a and...
Let R be the relation on the set of people given by aRb if a and b have at least one parent in common. Is R an equivalence relation? (Equivalence Relations and Partitions)
2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff...
2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements. 3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is...
Consider the relation R defined on the set Z as follows: ∀m, n ∈ Z, (m,...
Consider the relation R defined on the set Z as follows: ∀m, n ∈ Z, (m, n) ∈ R if and only if m + n = 2k for some integer k. For example, (3, 11) is in R because 3 + 11 = 14 = 2(7). (a) Is the relation reflexive? Prove or disprove. (b) Is the relation symmetric? Prove or disprove. (c) Is the relation transitive? Prove or disprove. (d) Is it an equivalence relation? Explain.
The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod...
The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod ? is an equivalence relation for an integer ? ≥ 2. a) What are the equivalence classes for R given a fixed integer ? ≥ 2? b) We denote the set of equivalence classes you found in (a) by Z_5. Even though elements of Z_5 are sets, it turns out that we can define addition and multiplication in the expected ways: [?] + [?]...
Let A = R x R, and let a relation S be defined as: “(x​1,​ y​1)​...
Let A = R x R, and let a relation S be defined as: “(x​1,​ y​1)​ S (x​2,​ y​2)​ ⬄ points (x​1,​ y​1)​ and (x​2,​ y​2)​are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you ​must​ give a counterexample.
Construct the 2 × 2 matrix for the linear transformations R 2 → R 2 defined...
Construct the 2 × 2 matrix for the linear transformations R 2 → R 2 defined by the following compositions. In each case, write down the matrix of each transformation, then multiply the matrices in the correct order. (a) A dilation by a factor of 4, then a reflection across the x-axis. (b) A counterclockwise rotation through π/2 , then a dilation by a factor of 1/2 . (c) A reflection about the line x = y, then a rotation...
Show that the relation 'a R b if and only if a−b is an even integer defined on the Z of integers is an equivalence relation.
Show that the relation 'a R b if and only if a−b is an even integer defined on the Z of integers is an equivalence relation.
Let S = {2 k : k ∈ Z}. Let R be a relation defined on...
Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT