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By Keith Devlin

Keith Devlin. you recognize him. you will have learn his columns in MAA on-line, you have heard him at the radio, and you have noticeable his well known arithmetic books. In among all these actions and his personal learn, he is been not easy at paintings revising units, features and good judgment, his standard-setting textual content that has smoothed the line to natural arithmetic for legions of undergraduate students.

Now in its 3rd version, Devlin has absolutely remodeled the e-book to mirror a brand new new release. The narrative is extra energetic and no more textbook-like. comments and asides hyperlink the subjects offered to the true global of scholars' event. The bankruptcy on complicated numbers and the dialogue of formal symbolic good judgment are long past in prefer of extra routines, and a brand new introductory bankruptcy at the nature of mathematics--one that motivates readers and units the degree for the demanding situations that lie forward.

Students crossing the bridge from calculus to better arithmetic desire and deserve all of the support they could get. units, features, and common sense, 3rd version is an inexpensive little e-book that each one of your transition-course scholars not just can find the money for, yet will really read…and enjoy…and research from.

About the Author

Dr. Keith Devlin is govt Director of Stanford University's middle for the examine of Language and data and a Consulting Professor of arithmetic at Stanford. He has written 23 books, one interactive booklet on CD-ROM, and over 70 released learn articles. he's a Fellow of the yankee organization for the development of technology, a global fiscal discussion board Fellow, and a former member of the Mathematical Sciences schooling Board of the nationwide Academy of Sciences,.

Dr. Devlin is usually one of many world's top popularizers of arithmetic. referred to as "The Math man" on NPR's Weekend version, he's a widespread contributor to different neighborhood and nationwide radio and television exhibits within the US and Britain, writes a per month column for the internet magazine MAA on-line, and frequently writes on arithmetic and desktops for the British newspaper The parent.

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Extra info for Sets, Functions and Logic: An introduction to abstract mathematics

Sample text

The sentence Vx¢(x) is true if and only if the truth set of ¢ is the entire universe U. The sentence 3x¢(x) is true if and only if the truth set of ¢ is not empty. As an illustration of these definitions, fix the set of all real numbers, 'fl, as the universe, and consider the following quantified sentences. • Vx(x > 3) : The truth set of the sentence x > 3 does not contain the number 2 (for instance), and hence is not the entire universe, 'fl, so the sentence Vx(x > 3) is false. • Vx(x i= 0) : The truth set of the sentence x i= 0 does not contain the number 0, and hence is not the entire universe, so the sentence Vx(x i= 0) is false.

Labl = -(ab) = (-a)b = lallbl. labl = ab = (-a)(-b) = lallbl. 3 Let a > O. Then for any z, Izl ~ a iff -a ~ z ~ a Proof: Exercise for the reader. e. 1, and the proof is complete. 0 If x, yEn, the positive number Ix-yl represents the distance between x and y on the real line. If x 1: y, then Ix - yl > O. Conversely, if Ix - yl > 0, then x 1: y. Hence x = y iff Ix - yl = O. By the triangle inequality, if X,y,z En, we have Ix - yl ~ Ix - zl + Iz - yl This inequality is also referred to as 'the triangle inequality'.

Vx(x+ 1> x) : The truth set for the sentence x+ 1> x is the entire set of reals, and so the sentence Vx(x + 1 > x) is true. • 3x(x > 3) : This sentence is true, since the truth set of the sentence x> 3 is not empty (it contains the number 71", for instance). • 3x( x + 1 > x) : This sentence is true since the truth set of x + 1 > x is not empty (being in fact the entire universe). • 3x(x 2 + 1 = 0) : This sentence is false, since the truth set is empty. (But notice that if the universe were not 'fl but C, the set of all complex numbers, then this sentence would be true!

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