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Introduction to Logic and Probability
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
What is a theorem? An implication whose truth value is true is called a theorem. In other words, it is a true statement of the form: if p then q where p and q are statements which are true or false but not both. How do we prove a theorem? One possible answer is the following: We must read and understand the proofs of other mathematicians. Then we can mimic their proof techniques to prove our theorem. In this brief section, we shall see some methods of proof often employed to prove theorems: proof by induction, direct proof, and indirect proof (or proof by contradiction).
Write Your Math Well
Published in Edward J. Rothwell, Michael J. Cloud, Engineering Writing by Design, 2017
Edward J. Rothwell, Michael J. Cloud
The basic idea of direct proof is to start with definitions, standard assumptions, and already established propositions, and work toward the desired conclusion. This is not the same as starting with the desired conclusion and working toward some other statement.
Write Your Math Well
Published in Edward J. Rothwell, Michael J. Cloud, Engineering Writing by Design, 2020
Edward J. Rothwell, Michael J. Cloud
The idea of direct proof is to start with definitions, standard assumptions, and proven propositions, and work toward the desired conclusion. This is not the same as starting with the desired conclusion and working toward some other statement.
Invariant output feedback stabilisability: the scalar case
Published in International Journal of Control, 2022
Aristotelis Yannakoudakis, Michael Sfakiotakis
By virtue of the analysis presented in the previous section, for any negative definite matrix , V is positive definite for every feedback gain k, making Hermite’s Bezoutian positive definite. By transforming to its Jordan form for a fixed k, it is easy to prove that the matrix L has the same zeros with Hermite’s Bezoutian. It loses rank when has zero or opposite eigenvalues. Hence, the zeros of are poles of V, and one can calculate critical gains directly from the determinant of V. But Proposition 2.2 applies to V, only if it has constant rank within critical intervals, a not obvious fact. The direct proof of Theorem 3.1 using Lyapunov’s method requires further investigation. Below, we highlight some of the interesting aspects of the Lyapunov analysis through application to the systems of Examples 1 and 2.
Characterizing introduction to proof courses: a survey of U.S. R1 and R2 course syllabi
Published in International Journal of Mathematical Education in Science and Technology, 2020
Within the Standard only ITP subcategory, which consists of 41% (72/176) of all ITP courses surveyed, 29 different texts were used, many of which are used for ITP courses in other categories. The most commonly used text, used by only 15% (11/72) of courses in this category, was Mathematical Proofs: A Transition to Advanced Mathematics, by Chartrand, Polimeni, and Zhang. Additionally, 8% (6/72) of Standard only ITP professors use their own lecture notes or notes developed by their department. As described earlier, these courses cover what we take as the standard set of topics as a means of introducing proofs and do not introduce any other mathematical topics. These topics are: symbolic logic, truth tables, logical connectives and quantifiers, negation, set theory, direct proof, proof by contradiction, and proof by induction, and basic functions and relations. The following examples illustrate topic lists typical of courses in this category. We provide excerpts from two Standard only ITP course syllabi: one which provides an overview of topics and another which gives a detailed week-by-week topic description (Figures 5 and 6).
On the estimation of the consensus rate of convergence in graphs with persistent interconnections
Published in International Journal of Control, 2018
Nilanjan Roy Chowdhury, Srikant Sukumar, Mohamed Maghenem, Antonio Loría
A simple inspection shows that γLP ≥ γM, that is, the method of proof in Loría and Panteley (2002) leads to a tighter estimate of the rate of convergence. However, the indisputable advantage of Theorem 3.4 is that it provides a strict Lyapunov function and a direct proof. This facilitates Lyapunov redesign as well as analysis of consensus of networked systems with drifts since, in contrast to the case of other statements, one does not need to rely on converse theorems.