# 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 1 Explanation:
 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 2 Explanation:
 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 3 Explanation:
 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 4 Explanation:
 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 5 Explanation:
 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
Question 6 Explanation:
There are 6 questions to complete.