query: How can $A \supset B$ be a synonymous of $A \Rightarrow B$?
positive_passages.docid: 152856
positive_passages.text: Using $p\supset q$ instead of $p\implies q$
negative_passages.docid: 46613
negative_passages.text: What's the difference between $\equiv$ and $\leftrightarrow$ in a formal proof?
negative_passages.docid: 143130
negative_passages.text: Proving that the supremum of $(a,b)$ is $b$.
negative_passages.docid: 86505
negative_passages.text: Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
negative_passages.docid: 69184
negative_passages.text: Proof that absolute value of supremum equals suprememum of absolute value
negative_passages.docid: 48614
negative_passages.text: Why is "$P \Rightarrow Q$" equivalent to "$\neg Q \Rightarrow \neg P$"?
negative_passages.docid: 68781
negative_passages.text: proving if $a \le b$ then $\sup(A) \le \inf(B)$
negative_passages.docid: 22297
negative_passages.text: Normal operator $\Rightarrow$ S invariant implies $S^\perp$ invariant
negative_passages.docid: 50142
negative_passages.text: If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$
negative_passages.docid: 148569
negative_passages.text: Embeddings $A → B → A$, but $A \not\cong B$?
negative_passages.docid: 93017
negative_passages.text: Isomorphism between $\Bbb R$ and $\Bbb R(X)$?
negative_passages.docid: 132532
negative_passages.text: Things around $(\forall x)(\Psi \Rightarrow \Phi) \Leftrightarrow ((\exists z)\Psi) \Rightarrow \Phi)$
negative_passages.docid: 28021
negative_passages.text: Showing that these two definitions of $\gcd(a,b)$ are equivalent
negative_passages.docid: 68449
negative_passages.text: $\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated
negative_passages.docid: 94425
negative_passages.text: What is $\sup(\sin(n))$?
negative_passages.docid: 99633
negative_passages.text: Let $α$ be an ordinal and $A$ be a set of ordinals. Then $\sup\limits_{β∈A} (α+β) = α+\sup\limits_{β∈A}(β)$
negative_passages.docid: 155884
negative_passages.text: What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?