Lattice

Question 1
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 _____.
A
0
B
1
C
2
D
3
GATE CSE 2017 SET-2   Discrete Mathematics
Question 2
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
A
P_{r} =0
B
P_{r} =1
C
0 \lt P_{r} \leq \frac{1}{5}
D
\frac{1}{5} \lt P_{r} \lt 1
GATE CSE 2015 SET-1   Discrete Mathematics
Question 3
Consider the following Hasse diagrams.
i.

ii.

iii.

iv.

Which all of the above represent a lattice?
A
(i) and (iv) only
B
(ii) and (iii) only
C
(iii) only
D
(i), (ii) and (iv) only
GATE IT 2008   Discrete Mathematics
Question 4
The following is the Hasse diagram of the poset [{a,b,c,d,e}, \prec ]

The poset is:
A
not a lattice
B
a lattice but not a distributive lattice
C
a distributive lattice but not a Boolean algebra
D
a Boolean algebra
GATE CSE 2005   Discrete Mathematics
Question 5
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?
A
{1}
B
{1}, {2, 3}
C
{1}, {1, 3}
D
{1}, {1, 3}, {1, 2, 3, 4}, {1, 2, 3, 5}
GATE CSE 2004   Discrete Mathematics
Question 6
In the lattice defined by the Hasse diagram given in following figure, how many complements does the element 'e' have?

A
2
B
3
C
0
D
1
GATE CSE 1997   Discrete Mathematics
There are 6 questions to complete.

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