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…

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

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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…

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

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Q: Let N be the set of all natural numbers and let R be a relation in N, defined by R = {(a, b): a is a…

A: Given  Set  = set of all natural numbers N Relation = R = { (a,b) : a is multiple of b }   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…

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Q: If P(A)=p, P(A|B)=q, P(B|A)=r and A,B are mutually exclusive then what is the relation between p,q…

A: Let A and B be two events which are mutually exclusive. The probability that event B occurs provided…

Q: Prove that the relation of congruence is an equivalence relation.

A: A relation is said to be an equivalence relation if and only if the relation is reflexive, symmetric…

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…

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Q: Which relation on the set {a, b, c, d} are equivalence relations and contain(i) (b, c) and (c, d)

A: The set {a, b, c, d}.

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

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Q: 3. On the relation R on the set {1, 2, 3, 4, 5} defined by the rule (x, y) e R if x+y<6. a) List the…

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

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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…

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Q: Describe the equivalence classes of the relation R = {(a, b) : a E Z,beZ, 13 divides (a - b)}

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

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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…

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

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

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

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

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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…

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