# Tagged Questions

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### $\forall$ At the beginning or at the end?

I have a set of real numbers $x_1, x_2, \ldots, x_n$ and two functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$. What are the differences between the following ...
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### Italic notation in logic

I have seen some books on formal logic where variables are written in italic, while statements are upright. Hence a statement could like like $\mathrm A(x_1, \ldots, x_n)$. How much of a standard is ...
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### Mathematical notation for expressing the top n elements

I would like to know what is the mathematical notation to express the top n elements. Look at the equation below. Here $x_w$ is a feature vector representing the contribution of a particular word $w$. ...
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### Notation for Model-Relation of formulae with free variables

Lets assume we have a formula $\mathsf{path}(x,y)$ with free variables $x,y$, and $\mathsf{acyclic}$ with no free variables on the signature $\tau = \{E\}$ (i.e. Graphs). Informally, what the formula ...
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### Representing a for loop with modulus in formal notation

I have the following section of code I am writing for research. Basically I need to formally represent the mathematical notation behind a set that follows: ...
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### Can I use “: ,” instead of “, implies” for this example?

I've to write this statement in a formal manner: if $x>1$ then $x^{2}>1$. Writing the result of the exercise I face this problem, I wonder if these two statements are equivalent: ...
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### Notation for the horizontal line found in sequent calculus for linear writing

Is there any known Unicode character which can be used as a substitute for the horizontal line which is normally used in texts presenting sequent calculus? I actually use the oblique bar ...
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### What does the dot symbol mean?

A = B ⇔ (∀x.x ∈ A ⇔ x ∈ B) What does "∀x.x" mean? This expression means: saying 2 sets are equal, is equivalent to saying they're the same set, right? Thank you.
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### Help with understanding the definition of operation

I'm having trouble understanding this excerpt from Wikipedia, which defines an operation: Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation ...
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### Velleman - How to prove it - Do these two statements really mean the same thing?

Hello and thanks in advance for reading! In How to Prove it P29 Velleman writes: " In general, the statement y ∈ { x | P(x)} means the same thing as P(y), ... " In my understanding the first ...
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### Are these statements in logic form correct?

Let M represent the set of all Mathematics courses and S represent the set of all students. Predicates: ...
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### Definitions of different kinds of recursive reducability.

So this is just a definition check. In recursion theory we have many kinds of reduction, according to my notes we have $\leq_1,\leq_m,\leq_{tt},\leq_{wtt},\leq_{TT},\leq_{T}$. Now I can't for the ...
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### How to read this logical statement in English?

Statement: ∀n ∈ Z, [(P(n) ∧¬(n=2)) ⇒ O(n)] where Z is a set of natural numbers P(n) is the predicate "n is a prime number" O(n) is the predicate "n is an odd number" I got this, but I don't think it ...
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### Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
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### How to read structural rules

In the wikipedia page on structural rules, we have the following "weakening" rule. $$\frac{\Gamma \vdash \Sigma}{\Gamma \vdash A, \Sigma}$$ This makes no sense to me. It seems to be saying that, if, ...
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### Is “iff” the same as equality if each member is a predicate?

"Iff" - if and only if ($\Leftrightarrow$ or $\leftrightarrow$, although the first usually carries a "meta" meaning, something that is not evaluated) - is used in $2x=3\Leftrightarrow x=\frac32$, ...
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$x^2=4 \Leftrightarrow x = 2 \lor x=-2$, but $x$ can't be equal to both $2$ and $-2$, unlike what the inclusive-or symbol ($\lor$) implies. Although it may seem obvious, it's just wrong to say that ...
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### Is there a symbol for a repeated logical and operation?

For example, given a predicate $P$ and a set of variables: $\{x_1,x_2,x_3,x_4,...\}$, is there a single symbol similar to $\sum_{k=1}^n$ or $\bigcap_{k=1}^n$ to denote the following? ...
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### Symbols: If but not only if

I was searching for a Latex symbol that indicates $A \Rightarrow B$ and $A \not\Leftarrow B$ ($B$ if not only if $A$, $B$ ifnf $A$). I thought of using $A \Leftrightarrow B$ with the left arrow tick ...
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### A few basic questions about the arithmetical hierarchy, mostly about terminology.

I was reading about the arithmetical hierarchy, and I have a few questions, mostly notational. For completeness, here's the definition given over at Wikipedia. The classifications $\Sigma_n$ and ...
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### Notation in propositional logic

If in propositional logic one is trying to simplify a formula by evaluating its subformula, would it be considered an abuse of notation to actually substitute the bits $\{0,1\}$ in for the formula, to ...
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### Symbol for “if and only if”: $\implies$ or $\iff$?

I was wondering about the iff sign in maths. I've never learned about it in school & see it a lot online. Usually the sign looks like this: $\implies$, but in math.stackexchange I always see this: ...
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### How do you define definition symbol :=?

How is "$:=$" defined formally and why? "$\iff$", "$=$", ...?
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### What is the conventional notation for these logic statements?

When I studied chemical engineering I often found the need to rewrite lecture notes, handouts and books in order to gain a thorough understanding of the subject I was reading. As much as time ...
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### Confusing symbol in papers on hybrid logic

In literature about hybrid logic I'm reading for my thesis I've come across the following symbol: ::= Now, I've never seen this notation before. I can also not ...
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### Using $p\supset q$ instead of $p\implies q$

I saw that a use for the notation $p\supset q$ instead of $p\implies q$ that got me a bit confused. One occurrences is in this Wikipedia link. It seems to me opposite than what it should be, let me ...
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### Is it a standard to say that $a \oplus a_{\small 1}=0$ or $a \veebar a_{\small 1}=0$?

I am trying to express the following: $a$ or $a_{\small 1}=0$ but only one of them equals zero. so if $a=0$ then $a_{\small 1}\neq 0$ and if $a\neq 0$ then $a_{\small 1}=0$. And I'm ...
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### Definition(s) for variable binding in first-order logic

The following statement made me realize that variable binding can be defined in first-order logic: The same holds for λ terms to define functions. There is no reason that they could not be ...
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### What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?

Sometimes reading on wikipedia or in this site (and in very different context like topology, arithmetic and logic) I have found these symbols $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$. They are ...
Regard this statement $x \ge 0$. According to my teacher, by negating this statement, it will become $x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?