Questions regarding the second half of the course are here.
A cord with knots. (Lecture).
Pythagoras (or one of his followers) stated the theorem in general and gave a proof for it. Implicitly it was well known already to the Egyptians and Babylonians. (Lecture).
Thales of Miletus. (Lecture)
The Chinese number system is superior: it is more systematic and there are less and shorter words for numbers. (Lecture).
It was mentioned because around that time people started to settle, thus having more time and more needs for mathematics. The last ice age was 12000-10000 B.C. (Lecture)
Two lenghts AB and CD are commensurable if they can be measured by the same unit. Formally: if there exists a rational number u and natural numbers x and y, such that u*x=AB and u*y= CD. (Lecture).
60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60); it is also related to early calendars (with 360 days); but, ultimately, we do not know. (Lecture).
An argument is deductively valid, if it is impossible for the premises to be true and for the conclusion to be false at the same time (Lecture; FOL, p.24).
An argument where the truth of the premises make the conclusion more likely to be true (Lecture).
(Handout #2: Weston, page 47).
A judgment is objective if its outcome does not depend on the judge; taste of coffee vs. validity of argument. (Lecture).
An argument is formal if the validity of the argument does not depend on the meaning of the symbols employed. (Handout #1).
not A From the assumption `not A' and the derivation of a contradiction you
infer `A'. (Lecture)
From the assumption `not A' and the derivation of a contradiction you infer `A'. (Lecture)
You know nothing, the argument could as well be invalid. (Lecture)
The argument is not valid. Because it is possible for the premises to be true, but the conclusion to be false, for example, if my income rises because of a raise and not because of the lowering of my taxes. (Lecture)
An argument is sound if it is valid and its premises are true (Lecture).
Propositions (statements, declarative sentences) (Lecture).
Number theory, set theory, utility theory, physics, and many more (Lecture).
If it is not possible to derive a contradiction from the axioms (Lecture).
An argument. (Lecture).
An axiom of a consistent axioms system is independent if its negation together with the remaining axioms is still consistent. Another way to say this is that the axiom cannot be proved from the other axioms. (Lecture).
Yes. One can present models for Non-Euclidean geometries. (Lecture).
In the 19th century: 1888 by Dedekind, and 1889 by Peano. (Lecture).
Yes in both cases. (Lecture).
1. 0 <= P(A) <= 1
3. P(A or B) = P(A)+P(B), when A and B cannot both be true at the same time.
(Lecture, Handout #10).
Conditional Probability: P(A|B) = P(A and B) / P(B)
Bayes' Theorem: P(A|B) = ( P(B|A)*P(A) ) / P(B)
(Lecture, Handout #10).
P(Thu) = 1/7
P( Thu | 80-110 ) = 0, there is no class on Thursdays.