Practice Problem Final MAT200

1. Consider the following statement: For any positive integer n, if n is divisible by 6, then n is divisible by 2. (a) Please use symbolic language to rewrite it. (b) Please write down its denial in the affirmative form. (c) Is the statement is correct or false? If yes, please prove it. If not, please disprove it. (d) Please write down its contrapositive statement. 2. Let A be the set of rational numbers which are greater than 2 and less than 7 and B be the set of all integers divisible by 3. (a) Use the condition and the symbolic language to rewrite A and B. (b) Please all elements in A \ B and compute |A \ B|. (Hint: Please write down the bijection. ) (c) Prove that if x 2 A, then x2 9x + 18 = 0 (d) Let C be a finite subset of (30, 1). If |C| = n, please compute |(A \ B) ⇥ C| and |(A \ B) [ C| Page 2 3. Let f (x) : R ! R be a function defined as follows: f (x) = ( x + a, when x 3 whenx  1 1 (a) Find the constant a so that f is a function from R to R. (b) Is f |(3,1) : (3, 1) ! R injection or not ? If yes, please prove it. If not, disprove it. (c) Justify whether f |(0,1) : (0, 1) ! R is an surjection or not. If yes, please prove it. If no, point out the image (you do not need to prove it). p (d) consider the function g(x) = 3 x : R ! R. Verify whether g is a bijection or not. If yes, please prove it and find out its inverse. If not, please explain the reason. Page 3 4. Let {an } be a sequence of real numbers satisfying that a1 = 1 an = 2an 1 (a) Use the inductive argument to prove that an = 2n (b) Recall that 1 X n=1 k X 1 for n 2 Z>0 an = a1 = ak + n=1 Use the inductive argument to prove that k 1 X k P n=1 Page 4 an n=1 an = (2k 1) for n 2 Z>0 5. Let {fn (x)} be a sequence of functions from R to R and f (x) be a function from R to R. The function f (x) is called the limit of the sequence {fn (x)} as n goes to infinity ( written as lim fn (x) = f (x)), if for any x 2 R and any ✏ > 0, there is a positive integer N such n!1 that for any integer n, we have that when n N , then |fn (x) f (x)|  ✏. (a) Use the symbolic language to rewrite it f (x) = lim fn (x) () n!1 (b) Write down its useful denial. f (x) 6= lim fn (x) () n!1 (c) Consider a sequence of functions fn (x) as follows: fn (x) = Please prove that lim fn (x) = 0 n!1 Page 5 x . n 6. Recall that a relation ⇠ over a set X is called an equivalent relation if it satisfies that for any x, y and z in X, we have that • x⇠x • x⇠y () y⇠x • x ⇠ y ^ y ⇠ z =) x ⇠ z Consider the relation ⇠ on R ⇥ (R {0}) as follows: For any (x1 , y1 ) and (x2 , y2 ) in R ⇥ (R {0}) (x1 , y1 ) ⇠ (x2 , y2 ) () x1 y2 x2 y1 = 0 (a) Prove that ⇠ is an equivalent relation on R ⇥ (R (b) Consider the function f : R ⇥ (R {0}). {0}) ! R f : R ⇥ (R {0}) ! R x (x, y) ! y Prove that f is surjection. (c) For any any (x1 , y1 ) and (x2 , y2 ) in R ⇥ (R {0}), prove that f (x1 , y1 ) = f (x2 , y2 ) if only if (x1 , y1 ) ⇠ (x2 , y2 ) (d) Use (c) to define the function f¯ from the equivalent class R ⇥ (R follows: {0})/ ⇠ to R as f¯ : R ⇥ (R {0})/ ⇠ ! R x [(x, y)] ! y where [(x, y)] is the equivalent class containing (x, y). Please use (b) and (c) to prove that f¯ is a bijection. (Hint: [(x1 , y1 )] = [(x2 , y2 )] in R ⇥ (R {0})/ ⇠ means that (x1 , y1 ) ⇠ (x2 , y2 ).) Page 6 7. Let x, y and z be three positive integers satisfying that 1 1 1 + + =1 x y z In the following, we will find all pairs (x, y, z) 2 Z3 satisfying the above equation. (a) Prove by contradiction that min{x, y, z}  3 In the following, we assume that 1 < x  y  z. Use (a) to have that 1 < x  3. That is, we have two cases: (1) x = 2 and (2). x = 3. (b) If x = 2, prove that (z 2)(y 2) = 4 (c) Use (b) to prove that if x = 2, then (y, z) = (4, 4) or (y, z) = (3, 6). (d) If x = 3, prove that (2y 3)(2y 3) = 9 (e) Use (c) to prove that if x = 3, then (y, z) = (3, 3). (Hint: z (f) Find out all pairs (x, y, z) 2 Z satisfying that 3 1 1 1 + + =1 x y z Page 7 y x)