Linked Questions

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0answers
54 views

Why does Spivak assert that $1\ne 0$? [duplicate]

Spivak, Calculus, prologue, page 6 says: Moreover, $1\ne 0$. (The assertion may seem a strange fact to list, but we have to list it, because there is no way it could possibly be proved on the ...
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0answers
32 views

Linear dependence over the null field [duplicate]

The set {0} with the + and × operations defined as 0+0=0 and 0×0=0 sarisfies the properties to be a field, so a vector space can be constructed over it, with the property $0x = x $ where x is a vector....
3
votes
2answers
1k views

Is the zero ring a domain?

Is the zero ring usually considered a domain or not? Wikipedia says: The zero ring is not an integral domain; this agrees with the fact that its zero ideal is not prime. Whether the zero ring is ...
3
votes
2answers
909 views

A complete graph has no vertex cut

In the Wiki: https://en.wikipedia.org/wiki/Connectivity_(graph_theory) It says: A complete graph with n vertices, denoted $K_n$, has no vertex cuts at all. Also, the node connectivity of a ...
5
votes
2answers
472 views

Are all existence proofs by contradiction?

Do all existence proofs proceed indirectly (a.k.a "by contradiction," "reductio ad absurdum," etc.)? For example, can we prove directly that in a field the additive identity element and the ...
3
votes
3answers
134 views

Prove for any field that $1\ne 0$ [closed]

Let $1$ be the multiplicative identity, so that $1\cdot a = a$ (where $a\in \mathbb{F})$. Let $0$ be the additive identity, so that $a+0=a$. Prove that $0\ne 1$. (Here we don't yet know that $0$ and $...
2
votes
2answers
124 views

Reasons behind the field axioms : $1 \ne 0$ and $1/0$ not defined

A very basic question about the axioms for multiplication in Rudin's "Principles of mathematical analysis, 3rd Ed": On page 5, we have the axioms (M1-M5) for multiplication for a field $F$: (M4) $F$ ...
0
votes
1answer
88 views

Axiomatics of Real Numbers - should $0\neq 1$ be considered as axiom?

I am analysing axiomatic approach to defining real numbers. There are two axioms that postulate existence of $0$ and $1$, namely (according to my notes): There exists an element $0\in\mathbb{R}$ such ...
0
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0answers
78 views

Prove using anxioms that $1 \neq 0$ - checking

Let $1=0$ we know that $x \cdot y=0$ if $x=0$ or $y=0$ so $1(1-1)=0 $ if $1=0$ or $1=1$ but $1=1$ is contradiction since we assume that $1=0$ so $1 \neq 0$ Is my prof correct ? If not how should it ...
1
vote
1answer
38 views

Soft question about integral domain

Here is a soft question that I am dealing with. Please tell me if it's correct or not. Suppose $A$ is a commutative ring with unity. Is $A$ a prime ideal of $A$? I think the answer is true, ...