“mcs”-2018/6/6一13：43一page6一#14 6 Chapter I What is a Proof? nonnegative numbers into prime numbers,unless it's a constant (see Problem 1.25). But this example highlights the point that,in general,you can't check a claim about an infinite set by checking a finite sample of its elements,no matter how large the sample. By the way,propositions like this about all numbers or all items of some kind are so common that there is a special notation for them.With this notation,Claim 1.1.3 would be Vn∈N.p(n)is prime. (1.2) Here the symbol V is read"for all."The symbol N stands for the set of nonnegative integers:0,1,2,3,...(ask your instructor for the complete list).The symbol "e" is read as“is a member of,”or“belongs to,.”or simply as“isin”The period after the N is just a separator between phrases. Here are two even more extreme examples: Conjecture.[Euler]The equation a4+b4+c4=d4 has no solution when a,b,c.d are positive integers. Euler (pronounced "oiler")conjectured this in 1769.But the conjecture was proved false 218 years later by Noam Elkies at a liberal arts school up Mass Ave. The solution he found was a 95800,b=217519,c =414560,d 422481. In logical notation,Euler's Conjecture could be written, va∈z+vbez+c∈z+Vd∈z+.a4+b4+c4≠d4 Here,Z is a symbol for the positive integers.Strings of V's like this are usually abbreviated for easier reading: a,b,c,d∈Z+.a4+b4+c4≠d4. Here's another claim which would be hard to falsify by sampling:the smallest possible x.y,z that satisfy the equality each have more than 1000 digits! False Claim.313(x3 +y3)=z3 has no solution when x,y,z Z. It's worth mentioning a couple of further famous propositions whose proofs were sought for centuries before finally being discovered: Proposition 1.1.4 (Four Color Theorem).Every map can be colored with 4 colors so that adjacent regions have different colors. 2Two regions are adjacent only when they share a boundary segment of positive length.They are not considered to be adjacent if their boundaries meet only at a few points.“mcs” — 2018/6/6 — 13:43 — page 6 — #14 Chapter 1 What is a Proof?6 nonnegative numbers into prime numbers, unless it’s a constant (see Problem 1.25). But this example highlights the point that, in general, you can’t check a claim about an infinite set by checking a finite sample of its elements, no matter how large the sample. By the way, propositions like this about all numbers or all items of some kind are so common that there is a special notation for them. With this notation, Claim 1.1.3 would be 8n 2 N: p.n/ is prime: (1.2) Here the symbol 8 is read “for all.” The symbol N stands for the set of nonnegative integers: 0, 1, 2, 3, . . . (ask your instructor for the complete list). The symbol “2” is read as “is a member of,” or “belongs to,” or simply as “is in.” The period after the N is just a separator between phrases. Here are two even more extreme examples: Conjecture. [Euler] The equation a 4 C b 4 C c 4 D d 4 has no solution when a; b; c; d are positive integers. Euler (pronounced “oiler”) conjectured this in 1769. But the conjecture was proved false 218 years later by Noam Elkies at a liberal arts school up Mass Ave. The solution he found was a D 95800; b D 217519; c D 414560; d D 422481. In logical notation, Euler’s Conjecture could be written, 8a 2 Z C 8b 2 Z C 8c 2 Z C 8d 2 Z C: a4 C b 4 C c 4 ¤ d 4 : Here, Z C is a symbol for the positive integers. Strings of 8’s like this are usually abbreviated for easier reading: 8a; b; c; d 2 Z C: a4 C b 4 C c 4 ¤ d 4 : Here’s another claim which would be hard to falsify by sampling: the smallest possible x; y; z that satisfy the equality each have more than 1000 digits! False Claim. 313.x3 C y 3 / D z 3 has no solution when x; y; z 2 Z C. It’s worth mentioning a couple of further famous propositions whose proofs were sought for centuries before finally being discovered: Proposition 1.1.4 (Four Color Theorem). Every map can be colored with 4 colors so that adjacent2 regions have different colors. 2Two regions are adjacent only when they share a boundary segment of positive length. They are not considered to be adjacent if their boundaries meet only at a few points