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Logical Design of Automation Circuits
Published in Stamatios Manesis, George Nikolakopoulos, Introduction to Industrial Automation, 2018
Stamatios Manesis, George Nikolakopoulos
Boolean algebra, as well as classical algebra, were founded on some basic postulates like those of commutative and distributive property of the two logical operations of AND and OR, the existence of the neutral elements 0 and 1, and many more. From these postulates, it is possible to extract a series of theorems that can be utilized in the simplification of the logical functions. In Figure 4.4, some of the most fundamental postulates and theorems are presented in graphical (contact symbol) form as well as in Boolean form. The profound logic behind these theorems is that they are not only useful in their mathematical form in the simplification of the logical functions (i.e., during the logical design that is going to be described subsequently), but they are also very useful in the empirical design of automation circuits.
Artifact Reduction by Post-Processing in Image Compression
Published in H.R. Wu, K.R. Rao, Digital Video Image Quality and Perceptual Coding, 2017
Using the distributive property of the DCT in matrix multiplication [SY00], i.e., DCT(XY)=DCT(X)DCT(Y),
Mathematical Foundations
Published in Chintan Patel, Nishant Doshi, Internet of Things Security, 2018
If a, b, c ∈ and operations defined over are ◻ and △, then operation (a◻b)△c = (a△c)◻(b△c) is satisfied. Then we can say that set satisfies the distributive property.
Understanding the properties of operations: a cross-cultural analysis
Published in International Journal of Mathematical Education in Science and Technology, 2021
Meixia Ding, Xiaobao Li, Ryan Hassler, Eli Barnett
The commutative, associative, and distributive properties undergird the arithmetic operations (NRC, 2001). Considering a, b, and c as any arbitrary numbers in a given number system, the commutative property of addition (CP+) states that a + b = b + a while the commutative property of multiplication (CP×) states that a × b = b × a. In other words, CP deals with the changing of order of numbers with the results invariant. Distinct from CP, the associative property of addition (AP+) states that (a + b)+ c = a +(b + c), while the associative property of multiplication (AP×) states that (a × b) × c = a × (b × c). As such, instead of changing the order of numbers, AP deals with changing the order of operations. Finally, the distributive property of multiplication over addition (DP) states that a × (b + c) = a × b + a × c, which involves the interaction between the two different operations.
Designing examples to create variation patterns in teaching
Published in International Journal of Mathematical Education in Science and Technology, 2018
Since the introduction of the Swedish Compulsory School, the description of mathematics in policy documents has moved in two different directions. On the one hand, the curriculum went from focusing on activities the students must participate in to describing the core content that they must learn. On the other hand, the standard of knowledge that the students must reach is described in terms of skills, which are always linked to a specific area of skills that could be interpersonal – for instance, collaboration – or skills that are related to content. In the specific context of skills within areas where the content is mathematics, worked examples are essential for providing the students with general tools that can be used to solve problems of various kinds [4]. Bills et al. [4] distinguish two attributes of good examples: transparency and generalizability. Transparency allows a learner to recognize an example as being generic (e.g. f(x) = (x − 1)2 + 1 is transparent with respect to the coordinates of the vertex), and generalizability allows a learner to identify an example's arbitrary and changeable features (e.g. f(x) = (x − a)2 + b). Zaslavsky [12] distinguishes among specific, semi-general and general examples, and identifies the following reasons for teachers' use of examples: conveying generality and invariance, explaining and justifying notations and conventions, establishing the status of students' conjectures and assertions, connecting mathematical concepts to real life experiences and the challenge of constructing examples with given constraints. For instance, a general example could be used to show the distributive property of multiplication over addition for algebraic expressions, a semi-general example could be used to show the distributive property for real numbers and a specific example could be used to calculate 5 × 23 via the distributive property.