Q. A partial order ≤ is defined on the set S = {x, b₁, b₂, … bₙ, y} as x ≤ bᵢ for all i and bᵢ ≤ y for all i, where n ≥ 1. The number of total orders on the set S which contain the partial order ≤ is ______

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Q. Let (A, ≤) be a partial order with two minimal elements a, b and a maximum element c. Let P:A –> {True, False} be a predicate defined on A. Suppose that P(a) = True, P(b) = False and P(a) ⇒ P(b) for all satisfying a ≤ b, where ⇒ stands for logical implication. Which of the following statements cannot be true?

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Q. Suppose a relation R = {(3, 3), (5, 5), (5, 3), (5, 5), (6, 6)} on S = {3, 5, 6}. Here R is known as _________

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Q. Consider the congruence 45≡3(mod 7). Find the set of equivalence class representatives.

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Q. Which of the following relations is the reflexive relation over the set {1, 2, 3, 4}?

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Q. Determine the partitions of the set {3, 4, 5, 6, 7} from the following subsets.

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Q. Determine the number of equivalence classes that can be described by the set {2, 4, 5}.

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Q. Determine the number of possible relations in an antisymmetric set with 19 elements.

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Q. For a, b ∈ Z define a | b to mean that a divides b is a relation which does not satisfy ___________

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Q. Which of the following is an equivalence relation on R, for a, b ∈ Z?

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Q. Determine the set of all integers a such that a ≡ 3 (mod 7) such that −21 ≤ x ≤ 21.

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Q. For a, b ∈ R define a = b to mean that |x| = |y|. If [x] is an equivalence relation in R. Find the equivalence relation for [17].

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