1. For (a, b) and (c, d) in R², define (a, b)R(c, d) by a² – c² = b2 - d². Determine whether the relation R in R2² is an equivalence relation or not. If it is not an equivalence relation, justify your answer. If it is an equivalence relation, provide a complete proof.

Q: Let x1, x2, y1, y2 be real numbers. The expression (x1, y1) ~ (x2, y2) means that x12 + y12 = x22 +…

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Q: 9.32. A relation Ris defined on the set A = {a+bv2: a,b € Q, a + bv/2 + 0} by x R y if æ/y E Q. Show…

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Q: Let A be the set of all students of a boys school. Show that the relation Rin A given by R = {(a, b)…

A: It is given that, A is the set of all students of a boys school.Show that R is an empty…

Q: 2. For each of the following relations, indicate whether it is a partial order, a strict order, or…

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Q: 6. Consider the relation R = {(a, b), (a,c), (c, b), (b,c)} on the set A = {a, b,c}. Which of the…

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Q: Either show the following relations on {0, 1, 2, 3} is an equivalence relation, or show  which…

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Q: (a) Ri (b) R2= {(1,2), (1,3),(1,4),(2,2),(3,3),(5,3),(5,4),(5,2)} /hich of them is an equivalence…

A: Any relation is equivalence relation if satisfy the following conditions where is set of elements a)…

Q: 4. For (1, yı) and (r2,y2) in R² we set (11. yı) ~ (12. Y2) →{+ =3+ yi. (a) Show that ~ defines an…

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Q: ' (a,b) ∼ (c,d) ⇔ a+d =b+c '' defined as '∼' relation is that a equivalence relation?

A: It is equivalence relation if (a, b) belongs to ZxZ

Q: Let R=}(x,y)|2 divides 8x+by} be a relation the Set H= {1, 2,3.4,5, 6. 10 Determine whether the…

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Q: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: Given the relation R= {(1,1), (2.2), (3.3)(4.4), (1.2). (2.1) (3.4). (4.3)) on the set X- (1,2.3.4).…

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Q: For each of the following relations, determine whether it is (a) reflexive, (b) symmetric, (c)…

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Q: Given a natural number c ∈ N. The relation Rc is defined as follows: ∀ a, b ∈ N :  (a, b) ∈ Rc ⇔ (∃…

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Q: For each of the following, prove that R is an equivalence relation.   Let (a,b), (c,d) ϵ R2. Define…

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Q: Given the relation R- {(1,1), (2.2). (3,3).(4.4), (1,2). (2,1). (3,4), (4,3)) on the set X…

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Q: Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by(i) R = {(a, b) : |a –…

A: Given: Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by(i) R = {(a, b)…

Q: Suppose A,S C U, (<) E LOrd(A), and B := (a1, b1), (a2, b2) E (A × B): D(S). If we define a relation…

A: Given A,S⊂A ≤ is the lattice of A. B is the power set of δ , P(S) so if b1,b2 be two subsets of B.…

Q: Let T be the relation defined by, For all a, b e Z, aTb iff a – b is odd. Is T an equivalence…

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Q: If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence…

A: Given, R1 and R2 are equivalence relations in a set A. So, both R1 and R2 are Reflexive, Symmetric…

Q: Let A 1, 2} and B = {1, 2, 3} and define a relation R from A to B as follows: Given any (x, y) E A x…

A: Given

Q: 1. Consider the relation R1: R - R such that (r1, y1)R1 (X2, y2) if and only if both points are on…

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Q: Define a relation C from R to R as follows: For any (x, y) ∈ R x R, (x, y) ∈ C means that x² + y² =…

A: (x,y) ∈ R × R (x,y) ∈ C means    x2 + y2 =4

Q: Characterize the relations in questions 14 and 15 in terms of whether each is reflexive,…

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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: Determine whether the following relations are equivalence relations. For those relations that are…

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Q: 9.15. A relation R is defined on Z by a Rbif |a – b| < 2. Which of the properties reflexive,…

A: Given : A relation is R is defined on Z is given by  a R b if a-b≤2  Which of the properties…

Q: Define the relation R on 72 = Z x Z by (a, b) R (c, d) if a + d = b+ c. Prove that R is ar…

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Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

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Q: 4. Let the relation R = { (a, b) e Z × Z l(a + b) = 1 (mod 5) }. a. Is the relation transitive?…

A: A relation R between two sets A and B is the set of all ordered pairs a,b such that a∈A and b∈B. We…

Q: 1. Consider the relation R1: R2 - R such that (r1, y1)R1 (x2, y2) if and only if both points are on…

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Q: Use a particular counterexample to explain why R fails to be an equivalence relation is the…

A: We take counter example   9R10,10R11⇒9R  11

Q: Determine, with justification, whether the following relations are equivalence relations. If so,…

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Q: Let S be the following relation on ℂ: S = {(x, y) ∈ ℂ2 : y − x is real}. Prove that S is an…

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Q: Consider the set Q = Z ×(Z 10), and the relation tupon QDefined as: (a, b) ↑ (c, d) e a·d=b·c a - d…

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Q: Which of the following definitions for the relation R (defined below) on A would constitute an…

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Q: For (a, b),(c,d) EN×N define the relation (a, b) ~ (b,c) if and only if ab = cd. Is this relation…

A: Explanation of the answer is as follows

Q: a) If R and S be two relations from set A to B. Show that (RNS)-¹ = R-¹ S-¹ b) Justify the following…

A: We know that ,  A relation R defined on a set A is called equivalence relation if R satisfies the…

Q: 2. Define a relation C from IR to IR as follows: For any (x, y) E IR x IR (x, y) EC means that x2 +…

A: We will use basic definitions of relation.

Q: For each of the following, prove that R is an equivalence relation Let (a,b), (c,d) ϵ R2. Define…

A: A relation R on a nonempty set X is an equivalence relation if it is (i) reflexive (ii) symmetric…

Q: Suppose there are two transitive relations R, S Disprove the claim that the composed relation S R…

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Q: Show that if for a relation R we know that R^2 ⊆ R, then R is transitive, and conversely.

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Q: Our universe U is Z (integers), and R is the relation defined by R(x,y):=x<y^2 For example,  R(3,-2)…

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Q: For universe S = R², define relation ~= as: (a,b) ~= (c,d) < 2a - b = 2c - d Prove that ~= is an…

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Q: Determine whether the given relation is an equivalent relation on {1, 2, 3, 4, 5} If the relation is…

A: To find: (i)the given relation is equivalence or not.             (ii) The equivalence class.

Q: Check the following relation R on the set A = {1,2, 3, 4} is reflexive, symmetric and or transitive…

A: Not reflexive.  Symmetric.  Transitive.  Not Equivalence relation.

Q: On the set {(a, b)} of all ordered pairs of positive integers, define (x1, Y1) ~ (x2, Y2) if and…

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Q: 1. Consider the following relations defined on the set {a, b, c}. For each relation, determine…

A: (a) Reflexive functions are the ones whose domain and range are the same  for instance a,a∈R1 So…

Q: Let S be a nonempty set. Consider the relation ~ on P(S) defined byA ~ B if A U BC = S. Is the…

A: Let S be a nonempty set. Consider the relation ~ on P(S)  defined by A~B if A∪Bc=S To show: "~" is…

Q: Either show the following relations on {0, 1, 2, 3} is an equivalence relation, or show which…

A: Either show the following relations on {0, 1, 2, 3} is an equivalence relation,