In this article, we present computerized proof techniques of Gosper, WilfZeilberger and Zeilberger that can be used for enhancing the teaching and learning of topics in discrete mathematics. We demonstrate by e
Penner, R. C.Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific Pub. Co., 1999. 19. Polya, G.How to Solve It: A New Aspect ofMathematical Methods. Ishi Press, 2009. 20. Rodgers, N.Learning to Reason: an Introduction to Logic, Sets and Relations. Wiley ...
The “results” (not all new) proved in this section are intended only to illustrate various proof techniques. Therefore, they are not labeled as “theorems.” Proof Technique: Direct Proof The most straightforward proof method is direct proof, a logical step-by-step argument concluding with the...
and relations. Additional features of the Second Edition include: An intense focus on the formal settings of proofs and their techniques, such as constructive proofs, proof by contradiction, and combinatorial proofs New sections on applications of elementary number theory, multidimensional induction, coun...
DiscreteMathematicsandItsApplication Logic:Proof 1 MethodsforProvingTheorems 2 Theorems,proofs,andrulesofinference Whenisamathematicalargument(or“proof”)correct?Whattechniquescanweusetoconstructamathematicalargument?Theorem–statementthatcanbeshowntobetrue.Axiomsorpostulatesorpremises–statementswhicharegivenandassumed...
theory to refer to proofs that make no use of complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques. ...
I’m teaching Discrete Mathematics this semester. It is our “intro to proofs” class. One of the proof techniques the students learn is proof by induction. I told the class that usually the base case for induction proofs are easy and that most of the work occurs in the inductive step. ...
Conditionals are ubiquitous in mathematics: we routinely express theorems usinguniversal conditionalsof the form ‘for allx, ifA(x)thenB(x)’. The logic of universal conditionals is underpinned by that ofpropositional conditionals, which take the form ‘ifA(x0)thenB(x0)’, wherex0is a specifi...
Herbst et al. (2010) have proposed that teachers might use the various functions of mathematical proof documented in the literature (e.g., verification, explanation, discovery, communication, systematisation, development of an empirical theory, and container of techniques) (de Villiers1990; Hanna an...
A partial solution has been provided by a collection of techniques which, although necessarily incomplete, have a high success rate when applied to the rewrite rule sets that arise in practical theorem proving. Each of these techniques involve defining a measure from terms to a well-founded set,...