GATE CSE 2017 Set 2 | Question: 21

Question

Consider the set  $X=\{a, b, c, d, e\}$  under 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 ______

Comments

https://www.geeksforgeeks.org/gate-gate-cs-2017-set-2-question-33/

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Already a lattice, in Hasse diagram we just omit directional arrows. And here they have included them.

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but there are two pairs which are missing (a,d) and (c,d) what about them?

Answer 1

The Hasse Diagram given is already a lattice having LUB and GUB. So, answer is 0.

Answer 2

In the given lattice, let's denote the elements as follows:

a is related to: a, b, c, d, e

b is related to: b, c, e

c is related to: c, e

d is related to: d, e

e is related to: e

So, if we analyze the relations:

a is related to itself and every other element in the lattice.

b is related to itself and some elements.

c is related to itself and fewer elements compared to b.

d is related to itself and even fewer elements compared to c.

e is only related to itself.

All pairs are indeed already present in the lattice. Therefore, the answer is zero.