57 Deductive Reasoning in Propositional Logic (q) (r) (s) (t) (u) 2.2.10 ¬P ∨ Q H P → Q P → Q H ¬P ∨ Q ¬(P ∨ Q) H ¬P ∧ ¬Q ¬P ∧ ¬Q H ¬(P ∨ Q) ¬(P ∧ Q) H ¬P ∨ ¬Q (v) ¬P ∨ ¬Q H ¬(P ∧ Q) (w) H (Q → P ) → ((Q → ¬P ) → ¬Q) (x) H (¬Q → P ) → ((¬Q → ¬P ) → Q) (y) If P H Q and P H ¬Q then H ¬P (z) If P, Q H R and P, ¬Q H R then P H R Using the Deduction Theorem, show that if any of ∧, ∨, or → is considered definable in terms of the others, the corresponding axioms in H can be derived from the others Euclid of Alexandria (c 325–265 BC) was the most prominent and influential mathematician of ancient Greece He wrote a monumental 13-book work, known as the Elements, in which he laid the systematic foundations of both geometry and arithmetic Elements encompassed the system of postulates and theorems for what is now called Euclidean geometry and is probably the most influential book written in the history of mathematics It was still used as the classic textbook for the study of geometry until the end of the 19th century, when David Hilbert revised and modernized it Furthermore, the book is written in a distinctly axiomatic style, so Euclid’s system of geometry can be regarded as the first axiomatic system of a mathematical theory The first book of Elements contains five postulates for points and lines in the plane The fifth postulate is the famous Parallel Postulate, stating that for every line and for every point that does not lie on that line there exists a unique line through the point that is parallel to the given line Since it was not as simple to state or as obvious as the other postulates, Euclid himself and many mathematicians after him tried for two millennia to derive it from the other four postulates However, all these attempts were futile It was only in the early 19th century that the mathematicians Gauss, Lobachevski, and Bolyai showed independently that negations of this postulate lead to the development of consistent, alternative Non-Euclidean geometries, such as elliptic and hyperbolic geometries The axiomatic approach of Euclid therefore turned out to be much more than just a matter of style and illustrated the great importance of the axiomatic method in mathematics, taken up later by Dedekind, Peano, Frege, Russell, Whitehead, Hilbert, Bernays, and their followers Besides geometry, Euclid also made fundamental contributions to number theory in Elements, including the study of prime numbers, the celebrated proof of existence of infinitely many prime numbers, Euclid’s lemma on factorization, and, of course, the Euclidean procedure for computing the greatest common divisor of two integers That procedure was one of the earliest and most famous instances of what is today called an algorithm See on the following page a copy of the first few definitions and postulates from an 1838 edition of Euclid’s Elements, published by Robert Simson in Philadelphia, USA