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Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram:
 
For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L3 = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ L3 chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then
  • a)
    Pr = 0
  • b)
    Pr = 1
  • c)
    0 < Pr ≤ 1/5
  • d)
    1/5 < Pr < 1
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Suppose L = {p, q, r, s, t} is a lattice represented by the following ...
Number of triplets in L3 = Number of ways in which we can choose 3 elements from 5 with repetition = 5 * 5 * 5 = 125.
Now, when we take x = t, then the given condition for L is satisfied for any y and z. Here, y and z can be taken in 5 * 5 = 25 ways.
Take x = r, y = p, z = p. Here also, the given condition is satisfied. So, pr > 25 / 125 > 1/5.
For x = q, y = r, z = s, the given condition is not satisfied as q ⋁ (r ⋀ s) = q ⋁ p = q, while (q ⋁ r) ⋀ (q ⋁ s) = t ⋀ t = t.
So, pr ≠ 1.
Hence D choice.
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Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram:For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L3= {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ L3chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Thena)Pr = 0b)Pr = 1c)0 < Pr ≤ 1/5d)1/5 < Pr < 1Correct answer is option 'D'. Can you explain this answer?
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