Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Line (geometry) wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
System of polynomial equations wikipedia , lookup
Elementary mathematics wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Recurrence relation wikipedia , lookup
Elementary algebra wikipedia , lookup
Partial differential equation wikipedia , lookup
Questions on Exercise 2.6: For all x there exists a y such that for all z, if z > y then z > x+y. If z > y then z > 5 + y. False. If z > y then z > 0 + y. True. Or x could be anything less than zero. Even for a specific y, (for all z, z > y implies z > x + y) means x ≤ 0. So effectively the statement is saying “for all x, x ≤ 0”. Which isn’t true. 1 For all x and for all y there exists z such that z > y implies z > x + y. This is true. The difference here is that the z is existentially quantified instead of universally, so you get to choose z. So you can choose z so that z ≤ y, which makes the implication true. Furthermore you can do this for any x and y. 2 Comments on homework An equation is not a statement, it’s part of a statement. The equation x2 + x − 2 = 0 could fit into a number of different statements. For example: The solutions to x2 + x − 2 = 0 are −2 and 1. The number 5 is not a solution to the equation x2 + x − 2 = 0. In the first statement, x is universally quantified: it says “for all x such that x2 + x − 2, x = −2 or x = 1. In the second, x is 3 not really a variable, it’s just another name for 5. Moral: Have to say what x is or quantify it some way or another. You can’t just have a free x floating around. In the case of the homework problem, for example, start your proof with “Let x be a solution to . . . .” There’s a difference between an if-then statement and it’s converse. “If x = 2 or x = −1, then x2 + x − 2 = 0” is not the same as “If x is a solution then x = 2 or x = −1.” and “If a = 0 or b = 0 then ab = 0. ” is not the same as “If ab = 0 then a = 0 or b = 0.” 4 The mechanical procedures for solving an equation can serve many different lines of mathematical reasoning. Your goal is to bring out those hidden lines. Example: For all real numbers y 6= 1, there exists a real number x such that (x + 1)/x = y. 10 -10 10 -10 5 Proof Let y be a real number not equal to 1, and set x = 1/(y − 1). Then 1 1 + (y − 1) x+1 y−1 + 1 = = = y. 1 x 1 y−1 This is an example of a proof which was constructed backwards. First we solved the equation for x, then stated the solution at the beginning of the proof. If you want to prove that a number satisfying an equation exists, the most direct way to do so is to exhibit the number up front. The work you did to solve the equation is not a necessary part of the proof. Once you have the number in hand, all you have to do is put it into the equation to show that it’s a solution. 6 Write converse and contrapositive of If it’s an apple, it’s red. Converse If it’s red then it’s an apple. Not an equivalent statement. Contrapositive If it’s not red then it’s not an apple. 7 Example Irrationality of Direct statement: If x = √ 2. √ 2, then x is irrational. Contrapositive: If x is rational, then x2 6= 2. The second formulation is easier because we have something to work with in starting the proof: “Suppose x is rational. Then x = a/b where a and b are integers (b 6= 0). etc.” 8 Find contrapositive and converse of • If xz = yz and z 6= 0, then x = y. Contrapositive: If x 6= y, then xz 6= yz or z = 0. Converse: If x = y, then xz = yz and z 6= 0. • If xy = 0, then x = 0 or y = 0. • If x 6= y then x2 6= y 2. Which is true and which false? 9 Proof by contrapositive: If 2n + 1 is a prime number then n is even. 10