Partial Order and Lattices for GATE

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Question 1

A partial order ≤ is defined on the set S= {x, a1, a2,.....an, y} as x < ai for all i and ai ≤ y for all i, where n≥1. The number of total orders on the set S which contain the partial order ≤ is

  • n!

  • n+2

  • n

  • 1

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Question 2

Let (5, ≤) be a partial order with two minimal elements a and b, and a maximum element c. 

Let P : S → {True, False} be a predicate defined on S.
Suppose that P(a) = True, P(b) = False and 
P(x) ⇒ P(y) for all x, y ∈ S satisfying x ≤ y, 
where ⇒ stands for logical implication.

Which of the following statements CANNOT be true ?

  • P(x) = True for all x ∈ S such that x ≠ b

  • P(x) = False for all x ∈ S such that x ≠ a and x ≠ c

  • P(x) = False for all x ∈ S such that b ≤ x and x ≠ c

  • P(x) = False for all x ∈ S such that a ≤ x and b ≤ x

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Question 3

A partial order P is defined on the set of natural numbers as follows. Here x/y 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)
Which of these ordered pairs of natural numbers are contained in P?

  • (i) and (iii)

  • (ii) and (iv)

  • (i) and (iv)

  • (iii) and (iv)

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Question 4

 

The minimum number of ordered pairs that need to be added to R to make (X, R) a lattice is _____.

  • 0

  • 1

  • 2

  • 3

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Question 5

Consider the following Hasse diagrams.

 

Which all of the above represent a lattice?

  • (i) and (iv) only

  • (ii) and (iii) only

  • (iii) only

  • (i), (ii) and (iv) only

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Question 6

The following is the Hasse diagram of the poset [Tex]\left[\{a,b,c,d,e\},≺\right][/Tex]
 

 

The poset is

  • not a lattice

  • a lattice but not a distributive lattice

  • a distributive lattice but not a Boolean algebra

  • a Boolean algebra

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Question 7

Which of the following statements is/are True for a group G?

  • If for all x, y ∈ G, (xy)² = x²y², then G is commutative.

  • If for all x ∈ G, x2 = 1, then G is commutative. Here, 1 is the identity element of G.

  • If the order of G is 2, then G is commutative.

  • If G is commutative, then a subgroup of G need not be commutative.

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Question 8

Given a set of values R={1,2,3,4,5,6,7}, What is the number of relations on this set which is both partial-order and equivalence relation?

  • 0

  • 1

  • 2

  • 3

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Question 9

Consider a partially ordered set (T, ≤) with two minimal elements, p and q, and a maximum element r. Let a predicate Q : T → {True, False} be defined on T such that:

  1. Q(p) = True
  2. Q(q) = False
  3. Q(u) ⇒ Q(v) for all u, v ∈ T such that u ≤ v, where ⇒ represents logical implication.

Which of the following statements CANNOT be true?

  • Q(x) = True for all x ∈ T such that x ≠ q.

  • Q(x) = False for all x ∈ T such that x ≠ p and x ≠ r.

  • Q(x) = False for all x ∈ T such that q ≤ x and x ≠ r.

  • Q(x) = False for all x ∈ T such that p ≤ x and q ≤ x.

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Question 10

Consider a partially ordered set (S, ≼) with two minimal elements, a and b, and a maximum element c. A predicate P : S → {True, False} is defined on S with the following rules:

  1. P(a) = True
  2. P(b) = False
  3. P(x) ⇒ P(y) for all x, y ∈ S such that x ≼ y, where ⇒ denotes logical implication.

Based on this, determine which of the following statements can be true:

  • P(s) = True for all s ∈ S such that s ≠ b.

  • P(s) = False for all s ∈ S such that s ≠ a and s ≠ c.

  • P(s) = False for all s ∈ S such that b ≼ s and s ≠ c.

  • P(s) = False for all s ∈ S such that a ≼ s and b ≼ s.

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