let A and B be sets with a maximum.
I want to know if:
$$ A \cap B $$
also has a maximum then.
I know that $$A \cap B $$
has a supremum and I already proofed that $$ sup(A \cap B) \le min\{supA,supB\} $$
Can someone help me? ;)
2026-04-07 16:23:02.1775578982
Does the intersection of two subsets (with a maximum) necessarily have a maximum too?
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You can take $A= (0,1) \cup (3,4]$ and $B= (0,2]$ as an counterexample. A has the maximum 4 and B has the maximum 2. The intersection is $A\cap B= (0,1)$ which doesn't attain its maximum.