An Invitation to Abstract Mathematics (Undergraduate Texts by Béla Bajnok

By Béla Bajnok

This undergraduate textbook is meant basically for a transition path into greater arithmetic, even though it is written with a broader viewers in brain. the guts and soul of this e-book is challenge fixing, the place every one challenge is thoroughly selected to elucidate an idea, reveal a method, or to enthuse. The workouts require particularly large arguments, inventive methods, or either, therefore delivering motivation for the reader. With a unified method of a various number of themes, this article issues out connections, similarities, and ameliorations between matters each time attainable. This e-book indicates scholars that arithmetic is a colourful and dynamic human company via together with historic views and notes at the giants of arithmetic, by means of declaring present job within the mathematical group, and through discussing many well-known and no more recognized questions that stay open for destiny mathematicians.

Ideally, this article will be used for a semester path, the place the 1st path has no must haves and the second one is a more difficult direction for math majors; but, the versatile constitution of the publication permits it for use in numerous settings, together with as a resource of varied independent-study and study tasks.

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1. Proof. 2n 1/ is a perfect number. 2 1/. 2n 1/; t and this is what we intended to prove. u 36 4 What’s True in Mathematics? , divisor, perfect number) and a sequence of true statements, and we mark the end of a proof by the symbol . When reading a proof, one needs to carefully verify that each statement in the proof is true. How can one be sure that each statement is true? ) This statement is clearly true for n D 1 (we get 1 D 1); we can also easily verify our statement for n D 2 (1 C 2 D 22 1), for n D 3 (1 C 2 C 4 D 23 1), etc.

Recall that each square of the chess board is colored by one of two colors (usually black or white) in an alternating pattern. This assures that each domino will cover exactly one square of each color. Thus, 31 nonoverlapping dominoes will cover exactly 31 black and 31 white squares. But the two diagonally opposite squares are of the same color, so, after removing them, we are left with 30 squares of one color and 32 of the other; therefore, the required tiling is not possible. The difficulty of this puzzle lies in the fact that we have to verify that something is impossible; this requires evaluating every possibility.

The area of the square equals Fn2 . Fn Fn 2 Fn 1 ✦ ✦ ✦✦ ✦ ✦ ✦✦ ✦ ✦ Fn ✦✦ ✦ ✦✦ ✂ Fn 2 Fn 1 ✂ ✂ ✂ Fn ✂ ✂ ✂ ✂ Fn 1 Fn 2 ✂ 2 1 Our next diagram is a rearrangement of the four regions so that they form a rectangle of side lengths FnC1 and Fn 1 , respectively. (Note that a segment of 30 3 How to Make a Statement? ) The area of the rectangle equals Fn 1 FnC1 . Fn 1 Fn Fn 1 1 and a segment of ✦ ✦✦ ✦ ✦ ✦✦ ✦ Fn 2 ✦✦ ✦✦ ✦ ✦ Fn ✦✦ ✦ ✦ Fn 2 ✦✦ ✦ ✦ ✦✦ Fn Fn 1 1 Since the rectangle of the second diagram must have the same area as the square of the first diagram, we get the claim Fn 1 FnC1 D Fn2 : (a) Find some counterexamples for this claim.