Suppose that A is a set, and fC Ax A is a relation on A such that f is symmetric f is a function Prove that f is a bijection of A with itself, that is, prove that f is both injective and surjective. (Hint: recall that a relation f C Ax A is a function iff (i) (Va e A)(3b € A)[(a, b) E f] (ii) if (a, b) E ƒ ^ (a, c) e f, then b= c.)

Q: (d) Let x be the complex number 1-i. Use de Moivre's Theorem to find the smallest ne N such that x"…

A: By De Moivre's Theorem,  (cos x + isinx )n = cos nx + isin nx By Euler's formula eix= cos x + isin x

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Q: 7. (6 points) Let T be a relation defined on the set A = {1,2,3} such that T = {(1,1), (1,2), (2,2),…

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A: Reflexive relation means every term of a set is related with itself

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Q: 1. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let P = P(A) be the power set of A. Let R be the relation on…

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A: Equivalence relation

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Q: 12. Suppose that B is a nonempty set, and let A denote the set of all functions from B to B. Let F…

A: B≠ϕ    A={f f:B→B function}    F={ f f:B→B bijective}    S={(f,g)∈A×A ∃h∈F s.t f=h-1∘g∘h} a…

Q: A = Z*; a R b if and only if |a – b| < 2.

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Q: 2. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Define a relation R on F…

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Q: (i) Let A = {1,2,3,4,5), then either find a relation on A that is not reflexive on A and not…

A: Solution:

Q: (a) x and y are in the same major. (b) x and y have the same last name. (c) x and y are taking a…

A: See the attachment

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A: Step 1:Part(a)Reflexive:To prove reflexivity, we need to show that for every function f ∈ F, fRf.…

Q: a. Let A= {0,1,2,3} and define a relation R on A as follows: R= {(0, 2), (0, 3), (2, 0), (2, 1)}. i.…

A: Since you have posted multipart question and we can answer three subparts. So, answering the first…

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A: Consider the relation  for  Reflexive: For   Thus,  is not reflexive. Symmetric: For  but…

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Q: 1. Let R be the relation on N defined by "aRb if EN or 5| (b- a)". So for example (6, 2) € R and…

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Q: Let A = {1, 2, 3, 4}. Give an example of a relation on A that is (R = Reflexive, S = Symmetric, T =…

A: (a) Consider the Relation          V = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3,…

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A: Given:

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

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