Gate 2004 Set theory question REVIEW

Question

https://gateoverflow.in/1067/gate2004-73

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?

1. {1}
2. {1},{2,3}
3. {1},{1,3}
4. {1},{1,3},{1,2,3,4},{1,2,3,5}

In this question why C) is not the answer.

Comments

That is what I am saying, its presence satisfies the given condition and its absence dissatisfies the given condition.

If we include {1} only then {1,3,5} $\wedge$ {1,2,4} = {1} means satisfy but for these two particular sets, but if we take {1,2,3} $\wedge$ {1,3,5} = {1,3}. Means if we do not include {1,3} then it dissatisfies the condition and its inclusive to the group will satisfy all set "$\wedge$" operations.

NOTE :- they are asking about complete lattice not lattice.

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You are not considering set relation correctly :

{1} $\subseteq$ {1,2} ,

what you are saying is for (1,2,3)  ∧ (1,3,5) = (1,3)

Relation is for subset not for common attribute.

Note has no ,meaning here it just said  lattice should be there .

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Thanks, @Anu007 I got your point.

GLB of two sets should be the subset of both and need not be the intersection.