Linked Questions

4
votes
2answers
305 views

Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”? [duplicate]

Possible Duplicate: Alternative ways to say “if and only if”? So when I come across mathematical definitions like "A function is continuous if...."A space is compact if....","Two ...
1
vote
2answers
204 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
votes
0answers
33 views

Shouldn't big oh definition be if and if only if, not just if? [duplicate]

This is from Discrete Mathematics and its Applications Shouldn't the if in that definition be an if and only if? Say we know that $n^2$ is in O($n^2$). Then from one side of the if and only if, we ...
113
votes
15answers
11k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
7
votes
5answers
11k views

Different ways to express If-Then

What are some different ways to write the conditional statement $p\implies q\,$, but in English? There's the obvious "If p, then q", but are there any other ways to write it? I'm looking for another ...
10
votes
5answers
11k views

Implication and equivalence arrows, when to use them?

In my course book we have something called implication arrows $\Rightarrow$ and equivalence arrows $\Leftrightarrow$ and I have never managed to understand them. When do I know which to use and how ...
10
votes
2answers
1k 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 ...
1
vote
3answers
2k views

proof: set countable iff there is a bijection

In class we had the following definiton of a countable set: A set $M$ is countable if there is a bijection between $\mathbb N$ and $M$. In our exam today, we had the following thesis given:If $A$ is ...
1
vote
2answers
928 views

Find the number of distinct equivalence classes $[f]$ of $R$.

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Define a relation $R$ on $\mathscr F$ by : for any $f,g \...
3
votes
2answers
160 views

deducibility from peano axioms

I have to solve the following problem: Using $\exists$ Introduction prove that PA$\vdash x\leq y \wedge y\leq z \longrightarrow x\leq z$: I used that if $x\leq y$ then $\ \exists \ r\ x+r=y$ and in ...
3
votes
2answers
118 views

Conventions on definitional if(f)

When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance: Definition 1: A set $A$ is open in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such ...