query: Why is $0 \neq 1$ included in this list of algebraic axioms?
positive_passages.docid: 116916
positive_passages.text: What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?
negative_passages.docid: 71653
negative_passages.text: Why is the Cartesian product of a set $A$ and empty set an empty set?
negative_passages.docid: 66451
negative_passages.text: In a ring, how do we prove that a * 0 = 0?
negative_passages.docid: 152297
negative_passages.text: Understanding quotients of $\mathbb{Q}[x]$
negative_passages.docid: 65768
negative_passages.text: What is the cardinality of the set of the empty set?
negative_passages.docid: 71604
negative_passages.text: Why is the determinant of the 0x0 matrix equal 1?
negative_passages.docid: 89535
negative_passages.text: fake proof of $\forall a. \forall b. a = b \to 1 = 0$
negative_passages.docid: 125161
negative_passages.text: Why is the Cartesian product of $A$ and $\varnothing$ empty?
negative_passages.docid: 35631
negative_passages.text: $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
negative_passages.docid: 22869
negative_passages.text: Prove $[0, 1]$ and $(0, 1]$ are equinumerous by use of bijection
negative_passages.docid: 32280
negative_passages.text: Meaning of $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$
negative_passages.docid: 67637
negative_passages.text: How come $\lambda(\mathbb{Q})= \inf\left\{\sum_{n \ge 1} \lambda([a_n,b_n)) : \mathbb{Q} \subset \bigcup_{n \ge 1} [a_n,b_n)\right\} = 0$?
negative_passages.docid: 7865
negative_passages.text: Complete Newbie: Axiom of Foundation Exercise
negative_passages.docid: 33705
negative_passages.text: Is $0$ a natural number?
negative_passages.docid: 35569
negative_passages.text: Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$
negative_passages.docid: 126862
negative_passages.text: Confused about $x \notin x$ example in Axiom of Specification
negative_passages.docid: 151356
negative_passages.text: Need Hints Prove "$((\neg \alpha \to \alpha) \to \alpha) $" Using Axiom 1,2,3 and MP and deduction theorem