Linked Questions

6 votes
11 answers
3k views

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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10 votes
2 answers
3k views

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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3 votes
1 answer
110 views

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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-1 votes
2 answers
868 views

$X$ 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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  • 85
0 votes
2 answers
264 views

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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  • 1,616
2 votes
2 answers
335 views

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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2 votes
0 answers
223 views

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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  • 29
0 votes
1 answer
138 views

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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  • 1,495
0 votes
0 answers
80 views

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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1 vote
1 answer
78 views

Should I use an equality sign or "iff" for a definition? [duplicate]

In many mathematical textbooks the definitions are given in the following form (considering even numbers as an example): "A natural number n is called even iff it is a multiple of 2". Now as ...
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  • 247
0 votes
0 answers
48 views

Does a mathematical definition have necessary and sufficient condition hidden in it? [duplicate]

Let us assume the following definition: `` S is said to be A if S satisfies the condition C. '' -----------(P) Can it mean that: `` S is A if and only if S satisfies the condition C. '' ---------- (...
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  • 1,333
0 votes
1 answer
36 views

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 ...
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0 votes
2 answers
38 views

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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0 votes
0 answers
34 views

I want clear a point about definitions. [duplicate]

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.
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  • 1,255
0 votes
0 answers
26 views

Could a metric space X be sequentially compact if one sequence in X has no convergent subsequence? [duplicate]

In Kreyszig compactness of metric space X is defined as : A metric space X is said to be compact if every sequence in X has a convergent subsequence. My confusion is simply because of the use of the ...
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  • 139

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