et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈ S such that a + c = b. Show that if R is a partial order, then (i) 0 ∈ S (ii) for any a, b ∈ S, a + b ∈ S. You do not need to write a formal proof – a clear, concise and correct justification is sufficient.

Q: Define the following infinite collection of subsets of the positive integers: A1 = {x|x E Z* and 0 <…

A: A relation R is said to be an equivalence relation, if it is  Reflexive : For every a∈R→(a,a)∈R…

Q: Is the following statement true or false? If R is any symmetric relation on a set A, then R is…

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Q: Answer the following question accordingly:

A: Given set is  B={1,2,3,4}

Q: (a) Give an example of a relation on the set {a, b, c, d] which is symmetric and transitive, but not…

A: (a) Given set is: a, b, c, d. To Do: We have to give an example of a relation on the given set a, b,…

Q: Let A = {-5,-4,-3,-2,-1,0,1,2,3} and define a relation R on A as follows: For all m, n ∈ A, m R n…

A: We know that For a given set A  and an equivalence relation R on A, the equivalence class of an…

Q: Let S ⊆ N, and for any c, d ∈ N, consider a relation R such that cRd if and only if there exists a ∈…

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Q: Determine whether or not the relation R on the set of all people is reflexive, netric,…

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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

Q: Define the relation R on Z as follows: for x, y ∈ Z, xRy if and only y−x ≥ 0. Determine whether R is…

A: partial order means reflexive and antisymmetric.

Q: Let A be a set and R = {(x, x) : x ∈ A} be the is-equal-to relation on A. Prove that a relation S on…

A: We are given any non-empty set A, on which a relation R = { (x, x) : x∈A } is defined.  Let S be any…

Q: Let A be the set of all permutations of the 26 letters in the English alphabet. The relation R on A…

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Q: Given the following relation on {1,2,3,4}, is this relation equivalence? R = {(1,1), (1,3), (1,4),…

A: An equivalence relation on a set is a binary relation that is reflexive, symmetric and transitive.

Q: 6. Consider the set A = 10 = {0, 1, 2, ..., 9} and consider the relation R on A determined by a Rb…

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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: Let R ⊆ ℝ×ℝ with R={(x,y)|⌈x⌉=⌈y⌉}, that is, x and y round up to the same number. Characterize R in…

A: Given problem is :

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A: The quadratic equation, y=ax2+bx+c can be expressed in the vertex form as y=a(x-h)2+k, where, the…

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: 17. Prove that if R is a symmetric, transitive relation on A and the domain of R is A, then R is…

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