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Probability Basics
Published in Arthur David Snider, Random Processes for Engineers, 2017
So do all infinite sets have the same cardinality? No. It is natural to call a set that can be matched up one to one with the positive integers “countable,” but the continuum of all real numbers in, say, the interval between 0 and 1 is not countable. To show this, identify each real number with its infinite decimal expansion:1/2 = 0.50000000000000….1/3 = 0.333333333333333….2/2=0.707106781186548….π/4 = 0.785398163397448….
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The term set is used to denote a collection, assemblage, or aggregate of objects. More precisely, a set is a plurality of objects that we treat as a single object. The objects that constitute a set are called the members or elements of the set. If a set contains a finite number of elements, we call it a finite set; if a set contains an infinity of elements, we call it an infinite set. A set that contains no elements at all is called an empty, void, or null set and is generally denoted ∅ $ \emptyset $ .
Natural Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
The set of real numbers has higher cardinality comparable to a countable set. In other words, the set of real numbers is a non-countable infinite set. It has a cardinality of ℵ1. An unsolved problem is that “Is there a set having cardinality strictly between that of a countable set and the set of real numbers?”.
A New Hybrid Method for Secure Data Transmission Using Watermarking based on Fuzzy Encryption in IoT
Published in IETE Journal of Research, 2023
Hossein Mohammadi, Abdulbaghi Ghaderzadeh, Amir Sheikhahmadi
Fuzzy set theory has been widely used to model the concepts of human thought and refers to the uncertainty in the available information for making decisions based on multiple criteria. The merit of the substitute against the criteria and the significant weight of the criteria in relation to the linguistic values expressed by the numbers are published. In the fuzzy set, linguistic variables are used to describe fuzzy conditions, converting linguistic variables into numerical variables and logical real values are replaced by unit intervals in the decision making process [18]. Therefore, mathematically, a set is defined as a finite, infinite, or count ably infinite set of elements. In each case, each element is a member of a set or not. However, in fuzzy systems, the element may be part of the set or outside of it. Therefore, the answer to the question: “X is a member of a set A” does not have a definite right or wrong answer. Figure 2 shows the block diagram of fuzzy logic.
The relationship between notions of infinity and strategies used to compare enumerable infinite sets
Published in International Journal of Mathematical Education in Science and Technology, 2022
Maryam Homaeinejad, Ali Barahmand, Asghar Seif
In this case, individuals encounter a concept, unlike their ordinary experiences in real life under the effect of finite sets. This type of notion implies one of the current strategies applied by individuals to compare enumerable infinite sets as the part-whole strategy. Various types of notions, however, lead individuals towards applying different strategies. According to some previous research (e.g. McDonald & Brown, 2008; Tsamir, 2001a; Tsamir & Dreyfus, 2002; Tsamir & Tirosh, 2007), most of the strategies used by students to compare enumerable infinite sets are as follows: Single infinity: all infinite sets are equivalent since there is only one infinity.Part-whole: all proper subsets of a given set have fewer elements than the set.Incomparability: comparing infinite sets is not possible and therefore these kinds of sets are incomparable.One-to-one correspondence: existence of a one-to-one correspondence between all elements of two sets indicates that they are equivalent.