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Elements of Algebra
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The existential quantifier “there exists” is denoted by ∃ while the symbol ∄ means “there does not exist.” The existential quantifier “there exists one and only one” or, equivalently, “there exists a unique” is denoted by ∃!. The universal quantifier “for all” or “for every” is denoted by ∀. The common symbol for the connective “such that” is ∋. Thus, the assertion ∀x∃y∋∀z:p(x,y,z) reads: “for every x there exists a y such that for all z, p(x,y,z) is true.”
Exploring Mathematical Statements
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Existential quantifiers are typically statements that claim the existence of (at least one) object of a certain type that satisfies additional properties. Examples include: Let A be a set, and let a∉A. There exists a subset B of A which contains a.There exists a number x that satisfies sin (x) = 2.There exists a number y that satisfies y2 < 0.
Fuzzy Sets and Mathematical Fuzzy Logic Basics
Published in Umberto Straccia, Foundations of Fuzzy Logic and Semantic Web Languages, 2016
Next we recap quantifier aggregations [242, 465, 466, 467]. Classical logic has two quantifiers, the universal ∀ and the existential ∃ quantifier. These are extremal ones between several other linguistic quantifiers such as most, few, about half, some, many, etc. Quantifiers can be seen as absolute of proportional (see, e.g., [242]). To what concerns us, we consider the proportional ones. In this case, a proportional type quantifier, such as most, can be represented as a fuzzy subset Q: [0,1] → [0,1] such that for each r ∈ [0,1], the membership grade Q(r) indicates the degree to which the proportion r satisfies the linguistic quantifier that Q represents. See Figure 8.7 for examples of fuzzy quantifiers.
Nested sequents for intermediate logics: the case of Gödel-Dummett logics
Published in Journal of Applied Non-Classical Logics, 2023
As usual, we say that the occurrence of a variable x in φ is free given that x does not occur within the scope of a quantifier. We say that y is free for x in φ when substituting y for x in φ does not cause y to become bound by a quantifier; e.g. y is free for x in , but not in . In addition, we let denote the substitution of the variable y for all free occurrences of the variable x in φ, possibly renaming bound variables to ensure that y is free for x in φ. We extend the definition of the complexity of a formula from the previous section with the following case: for .
Accounting for the variability of lecturing practices in situations of concept introduction
Published in International Journal of Mathematical Education in Science and Technology, 2022
In addition to the standard issues with quantifiers (the meaning of individual quantifiers, the meaning of the scope of a quantifier, the fact that ∀ and ∃ do not commute), with logical implication, and with the interpretation of as a distance, several specific issues have been documented regarding the (mis)understanding of this formal definition. Let us spell out two of them. First, studies show that students tend to regard the third quantifier as unnecessary (which echoes common informal formulations such as ‘comes as close to L as one pleases’). Second, the meaning of the first ∀ quantifier changes, depending on how the definition of convergence is used: In order to prove that a given sequence tends to a given number, any/all positive ϵ has to be taken into account, and the existence of at least one corresponding N-value has to be proved (we will call this ‘context 1’); by contrast, if some property is to be derived for a sequence which is known to converge, then one is at liberty to select and use one specific value for ϵ, and the existence of N is warranted by the definition (context 2). A basic understanding of the universal quantifier does not warrant the ability to shift viewpoints depending on the context.
Relationship between UAVs and Ambient objects with threat situational awareness through grid map-based ontology reasoning
Published in International Journal of Computers and Applications, 2022
Myung-Joong Jeon, Hyun-Kyu Park, Batselem Jagvaral, Hyung-Sik Yoon, Yun-Geun Kim, Young-Tack Park
In this manner, additional concepts for four elements can be expressed in terms of intersection (⊓), union (⊓), and negation (﹁), and regarding the quantifier, expressions such as ‘at least one individual’(∃) and ‘all individuals’ (∀) can be made. The expressions up to this point are ALC, and the added transitive role (R∈) is S. Here, H denotes the hierarchy of roles, O denotes nominals/singleton classes, I denotes inverse roles, and N denotes limiting the number of roles. Thus, SHOIN is an expression that contains all of the above, and SHOIN(D) denotes the addition of data. For this reason, OWL, which is a language that reflects the expressive power of SHOIN(D), is widely used not only on the web but also to implement applications. In this study, we constructed an ontology based on the expressive power of SHOIN(D) to conduct experiments.