Inf and sup of an intersection and union of 2 sets

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I'm looking for an example when there is no supremum of an intersection of 2 sets, also no supremum of an union (not the same example).

Each set needs to have a supremum, but not their intersection. Same goes for the union.

I'd really appreciate same example for infimum.

(I do undersand what all of the mentioned above are, I just struggle with finding a suitable example.)