query: Does the empty set have a supremum or infimum?
positive_passages.docid: 1307
positive_passages.text: Infimum and supremum of the empty set
negative_passages.docid: 7112
negative_passages.text: Product of sets and supremum
negative_passages.docid: 125317
negative_passages.text: Is this statement "Every element in the empty set is greater than itself." true or its negation is true?
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
negative_passages.docid: 63717
negative_passages.text: Is the closure of a set equal to the closure of the closure of that set?
negative_passages.docid: 5022
negative_passages.text: Can a set containing $0$ be purely imaginary?
negative_passages.docid: 125502
negative_passages.text: Empty set literal?
negative_passages.docid: 126633
negative_passages.text: Is a contradiction enough to prove a set equality to $\varnothing$?
negative_passages.docid: 121504
negative_passages.text: Supremum of a product measurable function...
negative_passages.docid: 86518
negative_passages.text: Does $X$ have a lower bound?
negative_passages.docid: 51687
negative_passages.text: Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty?
negative_passages.docid: 137511
negative_passages.text: Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
negative_passages.docid: 119180
negative_passages.text: How to prove that the set $A = \left\{ {p:{p^2} < 2,p \in {\Bbb Q^+}} \right\}$ has no greatest element?
negative_passages.docid: 99480
negative_passages.text: Why is predicate "all" as in all(SET) true if the SET is empty?
negative_passages.docid: 62208
negative_passages.text: If $A$ is a non-empty set, then $A \notin A$
negative_passages.docid: 123498
negative_passages.text: Is "empty set" an element of a set?
negative_passages.docid: 155999
negative_passages.text: How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$