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95 votes

Boolean search explained

The counting principle that applies here is inclusion-exclusion. $$ \left|X \cup Y\right| = \left|X\right| + \left|Y\right| - \left|X \cap Y \right|$$ To make the numbers work out, $\left|X \cap Y \...
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  • 1,012
63 votes
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Boolean search explained

Hint: The search x AND y will result in 10 000 hits.
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48 votes
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What exactly is the semantic difference between set and type?

To understand the difference between sets and types, ones has to go back to pre-mathematical ideas of "collection" and "construction", and see how sets and types mathematize these. ...
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38 votes

in O(n) time: Find greatest element in set where comparison is not transitive

The standard algorithm for finding a maximum still works. Start with $a_1$ and go over the elements, if you see a larger value, update the maximum to be that value. The reason this works is that every ...
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  • 13.2k
29 votes
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Recover a set with the information of the sums of all its subsets

No you can't. Consider any set $S=\{a,b,c\}$ with $a+b+c=0$, and the set $S'=\{a+b,b+c,c+a\}$. The subset sums for $S$ are $0, a, b, c, a+b, b+c, c+a, a+b+c=0$. The subset sums for $S'$ are $0, a+b, b+...
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  • 7,249
25 votes

in O(n) time: Find greatest element in set where comparison is not transitive

As Ariel notes, the standard maximum-finding algorithm given below: def find_maximum(a): m = a[0] for x in a: if x > m: m = x return m ...
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15 votes

What exactly is the semantic difference between category and set?

In brief, set theory is about membership while category theory is about structure-preserving transformations – but only about the relationships between those transformations. Set theory is only about ...
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  • 304
14 votes
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How to find the maximal set of elements $S$ of an array such that every element in $S$ is greater than or equal to the cardinality of $S$?

Sort T. Then take elements while T[i] >= i+1. For example sorted(T)=[6,4,3,3,1,1]. Then, ...
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13 votes

Boolean search explained

Document 1: The cat is on the table Document 2: My cat is black Document 3: The dog is under the table Document 4: What's the name of your cat? Document 5: This is a black and white photo Search for ...
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  • 12.2k
11 votes

What exactly is the semantic difference between set and type?

In practice, claiming that $x$ being of type $T$ usually is used to describe syntax, while claiming that $x$ is in set $S$ is usually used to indicate a semantic property. I will give some examples to ...
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  • 6,978
10 votes
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If $A \cap B$ or $A \cup B$ or $A \times B$ is recursively enumerable is it true to say that both $A$ and $B$ are recursively enumerable?

Notice that $\mathsf{RE}$ is not closed under complementation. Therefore there exists an $A\in\mathsf{RE}$ such that $B=A^c\notin\mathsf{RE}$, while $A\cap B$ and $A\cup B$ are trivially recursively ...
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  • 1,163
10 votes

What exactly is the semantic difference between set and type?

To start, sets and types aren't even in the same arena. Sets are the objects of a first-order theory, such as ZFC set theory. While types are like overgrown sorts. To put it a different way, a set ...
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10 votes
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Lambda Calculus as a branch of set theory

It's false. The $\lambda$-calculus arose through efforts to understand foundations of mathematics. Nowadays some people mistakenly equate foundations with set theory. The Stanford Encyclopaedia of ...
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9 votes
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Returning a random subset with length k of N strings while only storing at most k of them

Use reservoir sampling. This is a good description in Wikipedia, or in Knuth. Let's start with the simple case, where $k=1$. You always have one string in memory. When you read the first string, ...
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  • 140k
9 votes
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"Regular languages over a common alphabet are closed under union."

I see two possible points of confusion in your question, and I will address them separately. What is meant by the title of your post: ""Regular languages over a common alphabet are closed under union....
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  • 208
8 votes

Find a regular language that is "infinitely between" two other regular languages

Because regular languages are closed under complement and intersection, $L_2 - L_1$ is regular. Because it's also infinite, it contains words of arbitrarily large lengths. Therefore, by the pumping ...
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8 votes

How to find the maximal set of elements $S$ of an array such that every element in $S$ is greater than or equal to the cardinality of $S$?

From my comment originally: This is closely related to a quantity ubiquitous in academic productivity assessment, the Hirsh index, better known as the $h$-index. In short it is defined as the number ...
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  • 418
8 votes

Is the intersection of infinitely many recursive sets recursive?

For each $i\in \mathbb{N}$, take $S_i = \mathbb{N}\setminus \{i\}$. You can now build any set you want as $X = \bigcap_{i\notin X}S_i$. Similarly, an easier proof that a union of recursive sets can ...
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8 votes
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Is there a formal difference between $f:X \to X$ and $f\in X \to X$?

No, they're mostly notational variations. There are different connotations to the different notations, and different notations are common in different fields where they can mean quite different things....
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8 votes

How to get an element from an existential proposition in Type theory proof assistant (Lean prover)

The existential form of the axioms of set theory is convenient for the meta-theoretic explorations of set theory, such as forcing etc., where it is important to have a minimal language to worry about (...
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7 votes
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What does $\{$ a set $\}^{+}$ mean in the context of languages?

This is the Kleene plus. It stands for $$ L^+ = \bigcup_{i \geq 1} L^i. $$ Here $L^i$ is the set of concatenations of $i$ words from $L$. In words, $L^+$ consists of all concatenations of one or more ...
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7 votes
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Data structure for a static set of sets

Generically, these are sometimes called subset/containment dictionaries. The fact that you had partial matching in your question (but deleted it) is actually not a coincidence, because subset/...
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  • 1,230
7 votes
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Does "contains only" imply "contains"?

My view is that vernacular would consider that S is not empty, i.e. $\emptyset \neq S\subseteq T$, while mathematical language would consider that S can be empty, i.e. $\emptyset\subseteq S\subseteq T$...
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  • 19.1k
7 votes
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Number of finite strings over a countably infinite alphabet

It's countable. The set $S_\ell$ of strings of length $\ell$ is $\Sigma\times\dots\times\Sigma$, which is a finite product of countable sets, so is countable. Now, the set of all finite ...
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7 votes

Set theory pertaining to category theory and functional programming

The notation $f:E\times F \to G$ means that $f$ is a function that needs two arguments, one from $E$, one from $F$, and the image is in $G$. This is how the function $\text{Union}$ is defined: the two ...
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  • 7,127
6 votes

What is the point of (Compactness theorem in the) Overspill principle?

The theorem says that when a sentence has arbitrarily large (finite) models, then it also has infinite models. The antecedent of the theorem: $\phi$ is a sentence of predicate logic such that for ...
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  • 4,611
6 votes
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What is the name of the property where $f(A) \supseteq f(B)$ when $A\supseteq B$?

It is called monotonicity with respect to the inclusion ordering of sets. More precisely, it is in this case increasing monotonicity since the order is preserved. If the order was reversed, it would ...
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  • 19.1k
6 votes
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Finding a fixed-size set whose members are contained by the largest number of other sets

I believe your problem is a direct instance of the NP-hard Maximum Coverage Problem, which is related to Set Cover. From wikipedia, Maximum Coverage Problem: As input you are given several sets ...
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  • 76
6 votes
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The meaning of "set" in NP-complete problem

It doesn't matter. If a certain problem has one version in which the encoding of sets allows for repeated elements (which are ignored semantically), and another in which repeated elements are ...
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6 votes

How to find the maximal set of elements $S$ of an array such that every element in $S$ is greater than or equal to the cardinality of $S$?

There is nothing wrong with your algorithm, and of course most of recursive algorithms can be converted into loops, here a loop version of your recursive code: ...
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