4. Let A = {1,2, 3, 4, 5} and let S = P(A), the power set of A. For a, b e S, define aRb if a b. (a) (8 points) Either prove that R defines an equivalence relation on S, or explain why it does not. (b) (4 points) Is R antisymmetric?

Q: ) Let R = {(a,b) : 3keza = k * b} be a relation over the positive integers. Prove or disprove that R…

A: Given that R=(a,b):a=k*b To show reflexive it is enough to show that a~a. It is true because, a=1*a.…

Q: 1. Let A = Z+x Z. Let R be a relation defined on A as follows: For all (a, b), (c, d) € A, (a, b) R…

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Q: Let RC RX R with {(x, y) |x² =5 y²}. Characterize R in terms of whether it is reflexive,…

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Q: 4. (a) Let R be the set of all real numbers. Consider a relation S on R, i.e. SCRX R. which is…

A: Thank you

Q: 4. Show that the relation defined on Z as follows are equivalence relation (a) For all, n € Z, m R n…

A: Note that : Since you have posted  multiple questions, we will provide the solution only to the…

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Q: Let A = {1, 2, 3}. Give an example of a relation R on A such that Ris symmetric but is not…

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Q: 2. Define a relation R on Z as follows: a Rb if and only if 5 | (a²-b²). a) Prove that R is an…

A: A relation R on ℤ is defined by: a R b if and only if 5 | a2-b2 . We have to prove that R is an…

Q: Consider the relation R on R2 given by for any (a,b),(c,d)∈R 2, (a,b)R(c,d) if and only if a 2 +b…

A: According to the given information, Consider a relation R on R2 as:

Q: 4. Describe all Consider the relation R on R x R defined by (x1, y1) R(x2, y2) iff x₁ - 2y2 = x₂ -…

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Q: ) Let A = {1,2,3, 4} × {1,2, 3, 4}, and define a relation R on A by (1, Yı)R(x2, Y2) if r1 + y1 = x2…

A: Equivalence relation means it should satisfy reflexive, symmetric, Transitive conditions.…

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Q: Check if the following statement is TRUE or FALSE. Let R be the relation on R² defined by…

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Q: (1) Show v is an equivalence relation on S. (2) Describe the equivalence classes of . (3) Describe…

A: Given: Define ~ on S=a, b: a, b∈ℤ, b≠0 by a, b~c, d if and only if ad-bc=0. To show: 1) ~ is an…

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Q: 5. Let X be the set X = {1,2,3}, and let R be the relation defined on X by x₁ Rx₂ if x1 + x2 3. (a)…

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Q: Reflexive. • Anti-reflexive. • Symmetric. • Anti-symmetric. Justify all your answers. Transitive. •…

A: Let's consider a few examples of elements inside the relation R, where (a,b)∈ R if and only if…