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: [?] + [?] = [? + ?] and [?] ⋅ [?] = [??]
Construct the addition and multiplication tables for Z_4 and Z_5. Record sums and products in the
form [r], where 0 ≤ r ≤ 3 (or 4, respectively).
c) Let [?], [?] ∈ Z_10. If [?][?] = [0], does it follow that [?] = [0] or [?] = [0]?
d) How would you answer the question from (c) for Z_11, Z_12?, Z_13?
e) For which integers ? ≥ 2 is the following statement true?
“Let [?], [?] ∈ Z_5. If [?][?] = [0], then [?] = [0] or [?] = [0].”

Expert Solution

Colby Messinger answered 2 years ago

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:...

S = Z (integers), R = {(a,b) : a = b mod 5}. Is this relation...

S = Z (integers), R = {(a,b) : a = b mod 5}. Is this relation an equivalence relation on S? S = Z (integers), R = {(a,b) : a = b mod 3}. Is this relation an equivalence relation on S? If so, what are the equivalence classes?

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising...

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation. (c) (x1,y1)R(x2,y2) in R×R if x1∗y2 = x2∗y1.

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.

Consider the equivalence relation on Z defined by the prescription that all positive numbers are equivalent,...

Consider the equivalence relation on Z defined by the prescription that all positive numbers are equivalent, all negative numbers are equivalent, and 0 is only equivalent to itself. Let f ∶ Z → {a, b} be the function that maps all negative numbers to a and all non-negative numbers to b. Does there exist a function F ∶ X/∼→ {a, b} such that f = F ○ π? If so, describe it .

Prove that the equivalence classes of an equivalence relation form a partition of the domain of...

Prove that the equivalence classes of an equivalence relation form a partition of the domain of the relation. Namely, suppose ? be an equivalence relation on a set ? and define the equivalence class of an element ?∈? to be [?]?:={?∈?|???}. That is [?]?=?(?). Divide your proof into the following three peices: Prove that every partition block is nonempty and every element ? is in some block. Prove that if [?]?∩[?]?≠∅, then ???. Conclude that the sets [?]? for ?∈?...

Determine whether or not the following is an equivalence relation on N. If it is not,...

Determine whether or not the following is an equivalence relation on N. If it is not, determine which property fails. If it is, prove it. (a) x ∼ y if and only if x|y (x evenly divides y). (b) x ∼ y if and only if the least prime dividing x is the same as the least prime dividing y. In this case, instead if using N, use the set A = {2,3,4,5,...}. (c) x∼yifandonlyif|x−y|<5. (d) x∼yifandonlyifx=4. 2. Prove that...

For m, n in Z, define m ~ n if m (mod 7) = n (mod...

For m, n in Z, define m ~ n if m (mod 7) = n (mod 7). a. Show that -341 ~ 3194; that is to say 341 is related to 3194 under (mod 7) operation. b. How many equivalence classes of Z are there under the relation ~? c. Pick any class of part (b) and list its first 4 elements. d. What is the pairwise intersection of the classes of part (b)? e. What is the union of...

Given a natural number q ≥ 1, define a relation ∼ on the set Z by...

Given a natural number q ≥ 1, define a relation ∼ on the set Z by x ∼ y if x - y is divisible by q. (i) Show that ∼ is an equivalence relation. We will denote the set of equivalence classes defined by ∼ with Z=qZ. Also let x mod q denote the equivalence class to which an integer x belongs. (ii) Check that the operations (x (x mod q) + (y mod q) · (y mod q)...

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

Question