lecture notes
... proof by contradiction, vacuous proof, trivial proof, and proof by cases. We start with a direct proof. Such a proof shows, using the rule of inferences that we just learned, that if p is true, then q must be true. Any established mathematical fact proved before, axioms (facts assumed to be true at ...
... proof by contradiction, vacuous proof, trivial proof, and proof by cases. We start with a direct proof. Such a proof shows, using the rule of inferences that we just learned, that if p is true, then q must be true. Any established mathematical fact proved before, axioms (facts assumed to be true at ...
On the Question of Absolute Undecidability
... intractable that Luzin (1925) was led to conjecture that it is absolutely undecidable, saying that “one does not know and one will never know whether it holds”. Our first candidate for an absolutely undecidable statement is thus the statement PM that all projective sets of reals are Lebesgue measura ...
... intractable that Luzin (1925) was led to conjecture that it is absolutely undecidable, saying that “one does not know and one will never know whether it holds”. Our first candidate for an absolutely undecidable statement is thus the statement PM that all projective sets of reals are Lebesgue measura ...
Easyprove: a tool for teaching precise reasoning
... Every formula in the proof text is numbered, so that it can be referred to using this number both in the proof text and in the sidebar. Initially the proof has only one goal – the theorem one wants to prove – and no assumptions (the assumptions selected when creating a task are implicit and are not ...
... Every formula in the proof text is numbered, so that it can be referred to using this number both in the proof text and in the sidebar. Initially the proof has only one goal – the theorem one wants to prove – and no assumptions (the assumptions selected when creating a task are implicit and are not ...
Direct proof
... Sometimes, exhaustive proof isn’t an option, but we still need to examine multiple possibilities Example: Prove the triangle inequality. That is, if x and y are real numbers, then |x| + |y| ≥ |x + y|. Clearly, we can’t use exhaustive proof here since there are infinitely many real numbers to conside ...
... Sometimes, exhaustive proof isn’t an option, but we still need to examine multiple possibilities Example: Prove the triangle inequality. That is, if x and y are real numbers, then |x| + |y| ≥ |x + y|. Clearly, we can’t use exhaustive proof here since there are infinitely many real numbers to conside ...
Definability properties and the congruence closure
... that q(x, y) defines a linear order of cofinality in C, then L,o,o(Q~~ does not satisfy A-interpolation. It is not a Karp logic since it allows elementary classes of uncountable dense linear order so AL,oo~(Q~~ Lo~o~(Th).Compare with [MaSh2]. We have similar results for the logic Lo~,ot~ctr~d~ where ...
... that q(x, y) defines a linear order of cofinality in C, then L,o,o(Q~~ does not satisfy A-interpolation. It is not a Karp logic since it allows elementary classes of uncountable dense linear order so AL,oo~(Q~~ Lo~o~(Th).Compare with [MaSh2]. We have similar results for the logic Lo~,ot~ctr~d~ where ...