Consider the set ℕ× ℕ. We will informally think of the pair (x, y) ∈ ℕ× ℕ as representing the integer x − y. Of course, there are many different ways of representing the same integer. To fix this, we define a relation ∼ on ℕ × ℕ by (x, y) ∼ (n, m) if and only if x + m = n + y. a) Prove that ∼ is an equivalence relation. b) We shall abbreviate ℕ× ℕ / ∼ by the symbol ℤ*. Prove that the function f : ℤ* → ℤ given by f([x, y]) = x − y is well-defined and bijective.

Q: - Recall that for x, y € R, x - y = R is the absolute value of x-y. That is, if x-y 20 if x-y<0. x-y…

A:

Q: The relation on the real numbers R defined by a ~ b if both a > b and b>a is not an equivalence…

A:

Q: Let A = {1,2,3,4} and B = {5, 6, 7}. Let f be the relation f = {(1,5). (2,5), (3,6), (?, ?)} where…

A:

Q: 3. The DIIOwing relations are not equi properties that are not satisfied. Explain your answer. a. a…

A:

Q: The relation R on the set of all integer numbers is defined by R = {(x, y) = Z × Z: x - y ≤ 2} is an…

A: Given a relation R on the set of all integer numbers defined by R={(x, y)∈Z×Z:x-y≤2}.  We have to…

Q: 5. Identify each relation as linear, quadratic, or exponential. a) YA b) c) -4 -20 +2 -6 4 4 6 2 YA…

A:

Q: prove why each relation has or does not have the properties: reflexive, symmetric, anti-symmetric,…

A: The given relation R is as follows. R=x,y∈Z×Z:x is a multiple of y If x,y∈R, we denote x related to…

Q: 2 Explain why the relation on N defined as is not an equivalence relation. {(x,y): +y is a multiple…

A:

Q: 9. The relation ~ defined on the set of natural numbers by a ~ b means 'a – b is even is : A. an…

A: Equivalence relation : A relation which is reflexive, symmetric and transitive is an equivalence…

Q: 1. Verify that the relation ~ is an equivalence relation on the set S given. (a) S = R reals, a ~ b…

A:

Q: Decide whether each of the following relations is an equivalence relation or not.  {(a, b) | gpa of…

A: To prove equivalence relation, we have to prove that (i) the relation is reflexive, (ii) the…

Q: Consider following relations on the set of positive integer: R₁ = {(a,b) | a> b} R₂ = {(a,b) | a = b…

A: The objective of the question is to determine which of the given relations are symmetric and…

Q: 2. (p.214, #6) There are five different equivalence relations on the set A {a,b,c}. Describe them…

A:

Q: Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not…

A: 6 Given set is A=1,2,3. Consider the relation R and S on A is given by  R=1,1,2,2,3,31,2,2,1 And…

Q: Please answer the part (B). I have got a solution for the question on this platform but the answer…

A:

Q: Let be the relation defined on R by the condition that x - y if x³-y³ = 13(x - y). For example, 1-3,…

A:

Q: 3. Let X be a set and S and R equivalence relations on X. Prove or give a concrete counter example…

A: Since you have asked multiple questions, we will solve the first and second question for you. If you…

Q: Consider the relation on the set N x N given by (m, n)R(k, l) if m + n = k + l. List three elements…

A:

Q: Obtain the 16 relations of the set {0,1}, indicate which of these are reflexive, symmetric and / or…

A:

Q: 6. (a, b), C

A: Let R be the relation on the set of ordered pairs of positive integers such that a,b,c,d∈R if and…

Q: 7. Let A = {1,2,3,4}x{1,2,3,4}. Define an equivalence relation ~ by (x1,x2) ~ (x3,xa) iff xxx2 =…

A: Let A=1,2,3,4×1,2,3,4. The equivalence relation ~  is defined by,        "x1,x2~x3,x4 only if…