Linked Questions

Suppose that we are trying to define a mathematical object $M$. The statement of the definition generally takes the form (or some of its equivalent variant), A mathematical object is said to be $M$ ...

In Walter Rudin's Principles of Mathematical Analysis he defines open set as: "E is open if every point of E is an interior point of E." So this can be translated in logic as "If every point of E is ...

Are all definitions or axioms in logic biconditional (iff) statements? It would make sense to me that they would be. A lot of times I will read a definition though and it won't be written as an iff. ...

My textbook says "Let $X$ and $Y$ be sets. We say $X$ and $Y$ have the same cardinality if there is a bijection $f: X \to Y$." I was wondering why the text does not say "if and only if." A ...

There are topics with the same name but my question is not as abstract as in those. My question is as follows: taken a generic definition like $x\;\mathbf{ is\; something}$ if $y$ it could be written ...

I was wondering if definitions in mathematics are "if and only" statements? I know for sure that theorems are not "iff" statements. Thank you in advance for your help.

I often write my notes as logical statements and constantly wonder why people use only the "if" direction in the definitions instead of the "if and only if". Consider: "A homomorphism $\phi$ is said ...

An example from Rudin would be: (c) if $x + y = 0$ then $y = -x$. There may be times when one would have to use the fact that since $y = -x, x + y = 0$. While this is fairly intuitive, professors ...

My guess is that they say "iff", but I wanted to make sure as math books have never made it explicit. To take the field axioms for example, is it fair to say that it says both: $F_x\rightarrow A_x$...

I'm looking at the following definition The random variables $X_{1}, \ldots, X_{d}$ are said to be comonotonic if they admit as copula the Frechet upper bound. I am however not quite sure how to ...

Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence. This is how I'm given the definition of a compact metric space and it confuses me. ...

When reading books I pay attention that some of them giving a definition to some notion use "if" but others "if and only if". For example, A set is called empty if it has no elements A set is called ...

I want know wether "defnitions" are if and only if. For example if a set satsfies all four group axioms we say it is group but then we go other way also. thank you.

I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two. An explanation of why the example is a partial ...

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\...