Q. Let R₁ be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R₂ be another relation from B to C = {1, 2, 3, 4} as defined below:

i. An element a in A is related to an element b in B (under R₁) if a * b is divisible by 3.

ii. An element a in B is related to an element b in C (under R₂) if a * b is even but not divisible by 3. Which is the composite relation R₁R₂ from A to C?

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Q. The time complexity of computing the transitive closure of a binary relation on a set of n elements should be ________

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Q. Let A and B be two non-empty relations on a set S. Which of the following statements is false?

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Q. Determine the characteristics of the relation aRb if a² = b².

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Q. Let R be a relation between A and B. R is asymmetric if and only if ________

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Q. Let a set S = {2, 4, 8, 16, 32} and <= be the partial order defined by S <= R if a divides b. Number of edges in the Hasse diagram of is ______

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Q. The less-than relation, <, on a set of real numbers is ______

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Q. If the longest chain in a partial order is of length l, then the partial order can be written as _____ disjoint antichains.

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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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