Q. Suppose X = {a, b, c, d} and π1 is the partition of X, π₁ = {{a, b, c}, d}. The number of ordered pairs of the equivalence relations induced by __________

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Q. A partial order P is defined on the set of natural numbers as follows. Here a/b denotes integer division. i)(0, 0) ∊ P. ii)(a, b) ∊ P if and only if a % 10 ≤ b % 10 and (a/10, b/10) ∊ P. Consider the following ordered pairs:

i. (101, 22) ii. (22, 101) iii. (145, 265) iv. (0, 153)

The ordered pairs of natural numbers are contained in P are ______ and ______

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Q. The inclusion of ______ sets into R = {{1, 2}, {1, 2, 3}, {1, 3, 5}, {1, 2, 4}, {1, 2, 3, 4, 5}} is necessary and sufficient to make R a complete lattice under the partial order defined by set containment.

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Q. Consider the ordering relation a | b ⊆ N x N over natural numbers N such that a | b if there exists c belong to N such that a*c=b. Then ___________

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Q. Consider the set N* of finite sequences of natural numbers with a denoting that sequence a is a prefix of sequence b. Then, which of the following is true?

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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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