How to Prove It

A Structured Approach


Author: Daniel J. Velleman

Publisher: Cambridge University Press

ISBN: 1139450972

Category: Mathematics

Page: N.A

View: 5532

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

The Scars to Prove It


Author: Chris Gulino

Publisher: Xlibris Corporation

ISBN: 1483681491

Category: Poetry

Page: 298

View: 6570

CHRIS GULINO is a writer born in Buffalo, New York on C H R I S G U L I N O August 19, 1970 and now lives in Las Vegas, Nevada. He is a 43 year old Italian American who lives with his girlfriend Loretta Hembre and their soon to be born son Anthony Christopher Gulino. He enjoys playing drums and singing in his local Punk Rock band Urban Disposal and playing backgammon on his free time. Other than that, he is a very busy guy..

Crazy with the Papers to Prove it

Stories about the Most Unusual, Eccentric and Outlandish People I've Known in Four Decades as a Sports Journalist


Author: Dan Coughlin

Publisher: Gray & Company, Publishers

ISBN: 1598510681

Category: Biography & Autobiography

Page: 272

View: 908

Cleveland sports journalist Dan Coughlin shares stories of some of the more unique individuals and events he covered over the course of his forty-five-year career.

R&D is War- and I've Got the Scars to Prove It


Author: Clifford L. Spiro PhD


ISBN: 1300413247

Category: Business & Economics

Page: 184

View: 2259

The book consists of 25 vignettes describing industrial research projects, including successes and failures. The emphasis is not on the technology per se, but rather on the human and business side of R&D.

Appraising Lakatos

Mathematics, Methodology, and the Man


Author: György Kampis,L. Kvasz,Michael Stöltzner

Publisher: Springer Science & Business Media

ISBN: 9401707693

Category: Science

Page: 382

View: 7616

Imre Lakatos (1922-1974) was one of the protagonists in shaping the "new philosophy of science". More than 25 years after his untimely death, it is time for a critical re-evaluation of his ideas. His main theme of locating rationality within the scientific process appears even more compelling today, after many historical case studies have revealed the cultural and societal elements within scientific practices. Recently there has been, above all, an increasing interest in Lakatos' philosophy of mathematics, which emphasises heuristics and mathematical practice over logical justification. But suitable modifications of his approach are called for in order to make it applicable to modern axiomatised theories. Pioneering historical research in England and Hungary has unearthed hitherto unknown facts about Lakatos' personal life, his wartime activities and his involvement in the political developments of post-war Europe. From a communist activist committed to Györgyi Lukács' thinking, Lakatos developed into a staunch anti-Marxist who found his intellectual background in Popper's critical rationalism. The volume also publishes for the first time a part of his Debrecen Ph.D. thesis and it is concluded by a bibliography of his Hungarian writings.

Prolog and Natural-Language Analysis


Author: Fernando C. N. Pereira,Stuart M. Shieber

Publisher: Microtome Publishing

ISBN: 0971977704

Category: Computational linguistics

Page: 240

View: 3572

Give God The Praise


Author: Darrell Anthony Boone

Publisher: Xlibris Corporation

ISBN: 1465335641

Category: Poetry

Page: N.A

View: 4255

Chapter Seven New and Improved God saw all that he had made, and it was very good. and there was evening, and there was morning--the sixth day. By the seventh day God had finished the work he had been doing; so on the seventh day he rested from all his work. and God blessed the seventh day and made it holy, because on it he rested from all the work of creating that he had done. Genesis 2: 1-3Though you cannot know the totality of it, there are several conclusions you can draw from Scripture that give clues to understanding the Will of God:o God's Will is irrefutable. o God's Will is good. o God's Will is acceptable. o God's Will is perfect. o God's Will is already done. in order to gain insight into this you must follow the instructions of Jesus and look beyond the appearances to what is real. Do not conform any longer to the pattern of this world, but be transformed by the renewing of your mind. Then you will be able to test and approve what God's will is -- his good, pleasing and perfect will. - Romans 12:2To begin to discover what God's will for you is, let us go all the way back to the beginning. in Genesis the first chapter you find God busy preparing a place for you. On the First Day God created heaven & earth, and day & night (a unity of opposites which exists throughout all creation). in the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. and God said, Let there be light, and there was light. God saw that the light was good, and he separated the light from the darkness. God called the light day, and the darkness he called night. and there was evening, and there was morning--the first day. - Genesis 1: 1-5Light is wisdom; and the first day's work is the calling of light or wisdom into expression. Light represents intelligence, and darkness represents undeveloped capacity. Symbolically these are day and night. - Charles Fillmore, Mysteries of Genesis P.14On the Second Day God created the firmament, the expanse of sky, to separate the waters above (aspiration) from the waters below (desire). and God said, Let there be an expanse [firmament] between the waters to separate water from water. So God made the expanse and separated the water under the expanse from the water above it. and it was so. God called the expanse sky. and there was evening, and there was morning--the second day. - Genesis 1: 6-8The second step in creation is the development of faith or the firmament. the waters represent the un-established elements of the mind. ...unexpressed possibilities in mind. There must be a firm starting point or foundation established. This foundation or firmament is faith moving upon the unformed capacities of Spirit consciousness. the divine Logos--God as creative power--gives forth the edict Let there be a firmament. the first step or day in creation involves light or understanding, and the second step, faith in the knowing quality of mind. Mysteries of Genesis P.16 -17On the Third Day God divided the waters below into dry land and seas (the mineral kingdom) as a provision for the growth of the vegetable kingdom. and the land produced vegetation in bountiful supply, all with the means of reproducing and continuing the creation process. and God said, Let the water under the sky be gathered to one place, and let dry ground appear. and it was so. God called the dry ground land, and the gathered waters he called seas. and God saw that it was good. Then God said, Let the land produce vegetation: seed-bearing plants and trees on the land that bear fruit with seed in it, according to their various kinds. and it was so. the land produced vegetation: plants bearing seed according to their kinds and trees bearing fruit with seed in it according to their kinds. and God saw that it was good. and there was evening, and there was morning--the third day.- Genesis 1: 9-13The third step in creation is the beginning of the formative activity of the mind called imagination. This gathers the waters . . . together unto one place so that the dry land appears. Then the imagination begins a great multiplication of forms and shapes in the mind. Mysteries of Genesis P.18On the Fourth Day God created the sun (the greater light) the moon (the lesser light) and the stars. Thus was provided a source of Light and energy. All of this in preparation for what is to come next. and God said, Let there be lights in the expanse of the sky to separate the day from the night, and let them serve as signs to mark seasons and days and years, and let them be lights in the expanse of the sky to give light on the earth. and it was so. God made two great lights--the greater light to govern the day and the lesser light to govern the night. He also made the stars. God set them in the expanse of the sky to give light on the earth, to govern the day and the night, and to separate light from darkness. and God saw that it was good. and there was evening, and there was morning--the fourth day. - Genesis 1: 14-19The fourth step in creation is the development of the two great lights, the will and the understanding, or the sun (the spiritual I AM) and the moon (the intellect). These are but reflectors of the true light; for God had said, Let there be light: and there was light--before the sun and the moon were created. the firmament of heaven is the consciousness of Truth that has been formulated and established. Mysteries of Genesis P.19;On the Fifth Day the animal kingdom emerges as God calls forth the creatures of the sea and sky to feed upon the provisions of the sea and land. They too are blessed with all that is necessary to perpetuate creation as God commands them to be fruitful and multiply. and God said, Let the water teem with living creatures, and let birds fly above the earth across the expanse of the sky. So God created the great creatures of the sea and every living and moving thing with which the water teems, according to their kinds, and every winged bird according to its kind. and God saw that it was good. God blessed them and said, Be fruitful and increase in number and fill the water in the seas, and let the birds increase on the earth. and there was evening, and there was morning--the fifth day. - Genesis 1: 20-23The fifth step in creation is the bringing forth of sensation and discrimination. the creatures are thoughts. the birds . . . in the open firmament of heaven are ideas approaching spiritual understanding. Water represents the unformed substance of life, always present as a fecundating element in which ideas (living creatures) increase and multiply, just as the earth produces a crop when sown with seed. the birds represent the liberated thoughts or ideas of mind (heavens). Mysteries of Genesis P.22On the Sixth Day we see calling forth the creatures of the land to further develop the animal kingdom. and God said, Let the land produce living creatures according to their kinds: livestock, creatures that move along the ground, and wild animals, each according to its kind. and it was so. God made the wild animals according to their kinds, the livestock according to their kinds, and all the creatures that move along the ground according to their kinds. and God saw that it was good. Genesis 1: 24-25On the sixth day of creation ideas of life are set into activity. Cattle represent ideas of strength established in substance. Creeping things represent ideas of life that are more subtle in their expression, approaching closer to the realm of sense. They are the micro-organisms. the beasts stand for the free energies of life that relate themselves to sensation. Divine ideas are always instantly set into activity: and it was so. Mysteries of Genesis P.24Having created all of these things as a provision for God's final creati