Q1GATE 2017NAT

Consider the set X={a, b,c,d,e} under the partial ordering R={(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}. The Hasse diagram of the partial order (X, R) is shown below. The minimum number of ordered pairs that need to be added to R to make (X, R) a lattice is _____.

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Q2GATE 2015MCQ

Suppose L={p,q,r,s,t} is a lattice represented by the following Hasse diagram: For any $x,y\in L$ not necessarily distinct, $x\vee y$ and $x\wedge y$ are join and meet of x,y respectively. Let $L^{3}=\{(x,y,z):x,y,z\in L\}$ be the set of all ordered triplets of the elements of L. Let $P_{r}$ be the probability that an element $(x,y,z)\in L^{3}$ chosen equiprobably satisfies $x\vee (y \wedge z)=(x\vee y)\wedge (x\vee z)$ . Then

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📖 Explanation

The total number of ordered triplets $(x, y, z)$ from the set $L$ is $5^3 = 125$ . The given equation, $x\vee (y \wedge z)=(x\vee y)\wedge (x\vee z)$ , is the distributive property for a lattice.

The property does not hold for all triplets. For example, if we take $x=q, y=r, z=s$ :

$x \vee (y \wedge z) = q \vee (r \wedge s) = q \vee p = q$ .
$(x \vee y) \wedge (x \vee z) = (q \vee r) \wedge (q \vee s) = t \wedge t = t$ .
Since $q \neq t$ , the property fails, meaning the probability $P_r$ is less than 1.

However, the property holds for some triplets. For instance, if $x=t$ (the maximum element), the equation becomes $t \vee (...) = (t \vee ...) \wedge (t \vee ...)$ , which simplifies to $t = t \wedge t = t$ . This is always true. There are $1 \times 5 \times 5 = 25$ such triplets. Thus, $P_r \ge \frac{25}{125} = \frac{1}{5}$ . Since other combinations also satisfy the property, $P_r > \frac{1}{5}$ .

Combining these, we get $\frac{1}{5} < P_r < 1$ .

Q3GATE 2008MCQ

Consider the following Hasse diagrams. i. ii. iii. iv. Which all of the above represent a lattice?

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Q4GATE 2005MCQ

The following is the Hasse diagram of the poset [{a,b,c,d,e}, $\prec$ ] The poset is:

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Q5GATE 2004MCQ

The inclusion of which of the following sets into S = {{1, 2}, {1, 2, 3}, {1, 3, 5}, {1, 2, 4}, {1, 2, 3, 4, 5}} is necessary and sufficient to make S a complete lattice under the partial order defined by set containment?

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Q6MCQ

In the lattice defined by the Hasse diagram given in following figure, how many complements does the element 'e' have?

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