Math 1A Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Spring1A Find two or three classmates and a few feet of chalkboard Some logic 1 Prove that for every real number A there exists a real number B such that B A 2 Let be a relationship between numbers for any given real numbers a and b either a b is true or it s not For example might be or or or 6 or it might be the relationship that a b means a 3 a List a few more examples of relationships Is there a relationship that is always true Never true b If is a relationship let a b mean it is not true that a b Is a relationship c Give two examples of relationships so that a b if and only if b a d Is there a relationship so that EITHER a b OR b a BUT NOT BOTH If so give an example What about a relationship so that if a 6 b then EITHER a b OR b a e Give an example of a relationship so that For every real number a there exists a real number b such that a b Prove that your relationship has the desired property f Give an example of a relationship so that the following is FASLE For every real number a there exists a real number b such that a b Prove your example works 3 Let be a relationship Assume that for every real number a there exists a real number b which may depend on a such that a b Which of the following are NECESSARILY TRUE Which are NECESSARILY FALSE If a statement is neither necessarily true nor necessarily false give an example for which the statement is false and one for which it is true A statement is necessarily true if it follows from the assumptions in our case this means that any satisfying the assumptions will also satisfy any necessarily true statement There may be relationships satisfying the assumptions and also satisfying statements that are not necessarily true A statement is necessarily false if from the assumptions you can prove it is false no satisfying the assumptions also satisfies the statement a If a is a real number then a b for every real number b b If b is a real number then a b for every real number a c If a is a real number then there exists a real number b so that a b d If b is a real number then there exists a real number a so that a b e There exists a real number a so that for every real number b a b f There exists a real number b so that for every real number a a b 1 g There exists a real number a so that there exists a real number b with a b h There exists a real number b so that there exists a real number a with a b i There exists a real number a so that for every real number b it is not true that a b j There exists a real number b so that for every real number a it is not true that a b k For every real number a there exists a real number b so that it is not true that a b l For every real number b there exists a real number a so that it is not true that a b m There exist real numbers a and b so that it is not true that a b n For all real numbers a and b it is not true that a b 4 Let be a relationship How would you prove that For every real number a there exists a real number b such that a b is FALSE 5 Let be a relationship such that For every real number a there exists a real number b such that a b is FALSE Which of the statements in exercise 3 are NECESSARILY TRUE Which are NECESSARILY FALSE 6 Let a b be the following relationship If x a then x2 b Is it true that for any real number a there exists a real number b such that a b Prove your answers 7 Let a b be the following relationship If x2 a then x b Is it true that for any real number a there exists a real number b such that a b Prove your answer 8 Let and be two relationships How would you prove that a b implies a b How would you disprove it 9 Let and be two relationships How would you prove or disprove If x a then x b 10 Let and be two relationships Define by a b means If x a then x b a For a given pair a and b how would you prove a b b How would you prove that For any real number a there exists a real number b such that a b c How would you prove that For any real number b there exists a real number a such that a b d How would you prove that There exists a real number b such that for any real number a it is not true that a b e How would you prove that It is not true that for any real number b there exists a real number a such that a b 11 Let f be a function and K and L two numbers Let x a mean that 0 x K a Let x b mean that f x L b Let a b mean that x a implies x b What is another way of writing the following statement For any positive real number b there exists a positive real number a such that a b 12 In problem 11 let f x x2 1 and K 2 If L 5 is Statement true or false If L 3 is Statement true or false Prove your answers 2