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### Why every definition is an “iff”-type statement? [duplicate]

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 $M$ ...
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### If a set is open, does it mean that every point is an interior point? [duplicate]

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 ...
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### Definition of Equivalence Relation [duplicate]

I was going through the text "Discrete Mathematics and its Application" by Kenneth Rosen (5th Edition) where I am across the definition of equivalence relation and felt that it is one sided. ...
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### $X$ and $Y$ have the same cardinality if and only if there exist a bijection from $X$ to $Y$? [duplicate]

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 ...
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### Logic Definitions/Axioms (Are they iff statements?) [duplicate]

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. ...
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### Definition of “definition”: use iff or if? [duplicate]

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 ...
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### Are math definitions iff statements? [duplicate]

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.
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### Why do we use “if” in the definitions instead of “if and only if”? [duplicate]

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 ...
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### Why are definitions written as 'if-then' statements instead of 'if-and-only-if' [duplicate]

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 ...
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### Do axioms say “iff”? (or, is it “if”, or, “only if”). [duplicate]

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: $F_x\rightarrow A_x$...
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### Use of the word “if” in mathematical definitions [duplicate]

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 ...
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### A concern on the definition of compactness in a metric space [duplicate]

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. ...
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### Which one is a correct way to give definitions? [duplicate]

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'm experimenting with some different proofs and one of them related to modular arithmetic, so I have the following definition: We say that $a \equiv b \pmod k$ if there exists an integer $q$ such ...