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Introduction
Published in David V. Kalbaugh, Differential Equations for Engineers, 2017
A theorem is a mathematical statement that can be proven. The fundamental theorem of algebra states that polynomial equations of order n have n solutions. Another important theorem is that if the coefficients ai are all real then the solutions consist of real roots and roots that occur in complex conjugate pairs. That is, if x = α − iβ is a root of Equation 1.7 (α and β real) then so is x = α − iβ. Yet another important theorem is that if two polynomials of the same order are everywhere equal then their coefficients (a1,a2,...,an−1,an in Equation 1.7) are equal.
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).
Mathematical Proof
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
To set the scene more precisely, suppose that we are given a system of axioms, A1, A2, ..., An. A theorem is a statement about the terms of the system which is logically implied by the conjunction of the axioms. We can therefore define a theorem in the system formally as a proposition T such that (A1 ∧ A2 ∧ ··· ∧ An)⊢ T.
Practical wisdom in an age of computerisation
Published in Civil Engineering and Environmental Systems, 2020
Of course, we humans have shown much ingenuity in the past but there is a danger of a kind of 'technical triumphalism' that claims we can meet our challenges simply by carrying on as we have done. We need to understand that practical rigour is not the same as logical rigour. Rigour is the strict enforcement of rules to an end. Mathematical logic is the ultimate form of logical rigour: it has one value – truth. Theorems are deduced using axioms (rules) which are true-by-definition. Physical science aims at precise truth but truth as a correspondence to the facts, which have to be set in a context. Practical rigour is much more complex. It is meeting a need by setting clear objectives involving many values (some in conflict) and reaching those objectives in a demonstrably dependable and justifiable way. Practical rigour is about using dependable evidence to move a process towards a specific goal. That is why the Italian Flag of evidence is so important to the Iopm. Practical rigour implies practical intelligence, which in turn implies practical experience. In other words, experience is necessary but not sufficient for practical intelligence – a capacity to learn, reason and understand practical matters. And practical intelligence is necessary but not sufficient for practical rigour. That is because practical intelligence and rigour require reflective learning and development on that experience.
Undergraduates’ propositional knowledge and proof schemes regarding differentiability and integrability concepts
Published in International Journal of Mathematical Education in Science and Technology, 2018
In analysis, the proposition that can be proven to be true every time is called a theorem. The teaching content of the analysis course frequently comprises the converse statements of the theorems, either proving or disproving the statements. The students also had various difficulties with the process of proving or disproving these propositions [9,19]. Some studies that underlined the epistemological origin of the difficulties encountered in analysis emphasized that the students do not have enough knowledge and awareness regarding mathematical logic [20,21]. Various expressions in analysis require propositional logic and logical reasoning through use of bi-directional perspective. Additionally, many theorems related to continuity, differentiability, and integrability are presented together with conditional statements in this context. For instance, Tall [21] stated that the students who could not understand the logic behind a statement had difficulty understanding and interpreting the proofs. Selden & Selden [20] also observed that the students who lack an understanding of the meaning of quantifiers in theorems in analysis also made logical errors. Compound propositions in the type of converse, inverse, or contrapositive are prepared with the use of conditional statement forms or quantifiers (such as ‘if …, then …’, ‘only if’, ‘if and only if’ ‘,∀’ or ‘∃’), and these propositions in analysis require reverse thinking. For instance, the following theorem under the title of ‘Differentiability Implies Continuity’, has been given coverage in many calculus textbooks: if f is differentiable at c then f is continuous at c (for every function f:ℜ → ℜ and c ∈ ℜ). This true proposition shows that all differentiable functions are also continuous at the defined point [22]. It also has the following interpretation: if a function has a discontinuity at a point (for instance, a jump discontinuity), then it cannot be differentiable there. If we think in the reverse direction, continuity does not guarantee differentiability. Therefore, the converse of this proposition is not true.1Moreover, textbooks also cover the proof or disproof of propositions containing the binary relations between differentiability and integrability, and also continuity and integrability [1,2,17,22,23]. While a counterexample is given for false propositions, direct or indirect proof processes are followed in true propositions. The students should have knowledge of propositional logic to successfully interpret the theorems in analysis [20,21,24].