Linked Questions
40 questions linked to/from Are "if" and "iff" interchangeable in definitions?
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 $M$ ...
user170039
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 ...
user462561
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.
...
- 727
-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 ...
- 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. ...
- 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 ...
- 2,629
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 ...
- 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.
- 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:
$F_x\rightarrow A_x$...
- 111
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 ...
- 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. '' ---------- (...
- 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 ...
- 978
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. ...
user346936
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.
- 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 ...
- 139