Let P be the partial order defined on the set {1, 2, 3, 4} as follows: P = {(x, x) | x in {1, 2, 3,

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GATE-CS is a competitive examination conducted in India. This question is from the 2024 GATE-CS Discrete Mathematics paper.

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Question

Let P be the partial order defined on the set {1, 2, 3, 4} as follows: P = {(x, x) | x in {1, 2, 3, 4}} union {(1, 2), (3, 2), (3, 4)}. The number of total orders on {1, 2, 3, 4} that contain P is __________

Accepted Answer

Answer: 5Concept -- Partial Orders, Linear Extensions, and Total Orders: The partial order P on {1, 2, 3, 4} has the following constraints (beyond reflexivity): 1 The constraints are: 1 before 2, 3 before 2, and 3 before 4. Element 2 must come after both 1 and 3. Element 4 must come after 3. No constraint between 1 and 3, no constraint between 1 and 4, no constraint between 2 and 4 (beyond 3 before 2 and 3 before 4).We enumerate valid permutations by placing elements in positions 1 to 4 left to right:3 must precede both 2 and 4. 1 must precede 2. Possible placements:3, 1, 2, 4 -- valid (3 before 2, 3 before 4, 1 before 2)3, 1, 4, 2 -- valid3, 4, 1, 2 -- valid1, 3, 2, 4 -- valid1, 3, 4, 2 -- validChecking others: 1, 4, 3, 2 -- 3 before 2 (yes), 3 before 4? No, 4 comes before 3. Invalid. By systematic enumeration, exactly 5 linear extensions exist.

Let P be the partial order defined on the set {1, 2, 3, 4} as follows: P = {(x, x) | x in {1, 2, 3, 4}} union {(1, 2), (3, 2), (3, 4)}. The number of total orders on {1, 2, 3, 4} that contain P is __________

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Let P be the partial order defined on the set {1, 2, 3, 4} as follows: P = {(x, x) | x in {1, 2, 3, 4}} union {(1, 2), (3, 2), (3, 4)}. The number of total orders on {1, 2, 3, 4} that contain P is __________

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