Numbers 6 and 7 on the attachment below please .......

 

 

Math 109. Instructor: Chow
Homework #2. Due 3:00 pm on Monday, July 11, 2016.
REVISED
Problem 1: Prove, using logical argument from the de…nitions, that
A [ (B \ C) = (A [ B) \ (A [ C) :
That is, prove that x 2 A [ (B \ C) if and only if x 2 (A [ B) \ (A [ C) : Hint: You may use the fact that: ‘P or (Q and R)’is logically equivalent
to ‘(P or Q) and (P or R)’.
Problem 2:
(i) Prove: 8x 2 Z+ , 9y 2 Z+ , y > 3x + 2. (ii) Prove: 8x 2 Z, 9y 2 Z, y > x2 + 9. (iii) DISProve: 9x 2 R, 8y 2 R, y < x2 .
(iv) Disprove: 8n 2 Z+ , 9m 2 Z+ , n m2 n + 39. Problem 3:
(i) Prove: 9x 2 R, 8y 2 R, 2xy = x3 + 2x2 .
(ii) Disprove: 9x 2 R, 8y 2 R, 2xy > 1. (iii) Prove: 8x 2 R, 9y 2 R, x2 + 1 y = x3 + 2x2 .
(iv) Prove: 8x 2 R Q, 9y 2 R
Problem 4: Let n 2 Z. Q, xy = 1. Let P (n) be the statement: There exists q 2 Z such that n = 5q + 3: Let Q(n) be the statement: There exists p 2 Z such that n2 = 5p + 4: Prove that P (n) implies Q(n): Hint: This is a direct argument, not a proof by induction.
Problem 5: Let f : [a; b] ! R be a di¤erentiable function. The mean
value theorem says that there exists c 2 (a; b) such that f 0 (c) = f (b)b af (a) .
Use the mean value theorem to prove that the function f : R ! R de…ned by
f (x) = 13 x3 + 3x2 + 10x is strictly increasing. (Recall that strictly increasing
means that for any a < b, f (a) < f (b).)
Problem 6: We say that limx!1 f (x) = 1 if for any M 2 R there
exists N 2 R such that if x N , then f (x) M .
The intermediate value theorem says that if f : [a; b] ! R is a continuous
function and if y is between f (a) and f (b), then there exists x 2 (a; b) such
that f (x) = y.
1 (a) De…ne, analogously to the above, what it means for limx! 1 f (x) =
1.
(b) Let f : R ! R be a continuous function with limx! 1 f (x) = 1
and limx!1 f (x) = 1. Prove: If y 2 R, then there exists x 2 R such that
f (x) = y.
Problem 7: Let I denote the irrational numbers. De…ne the function
f :I I!R R by
f (x; y) = x + y; x2 + y 2 :
(i) Does there exist (x; y) 2 I (ii) Does there exist (x; y) 2 I I such that f (x; y) 2 I I such that f (x; y) = Q f1g?
I? Problem 8: Given n 2 N, let Nn = f1; 2; : : : ; ng = fa 2 Z j 1 a ng.
Let X be a …nite set. The number of elements in X, called the cardinality of
X, is denoted by jXj. We have jXj = n if and only if there exists a bijection
f : Nn ! X. Answer correctly the following (no need to prove anything).
(i) If X Y , then how are jXj and jY j related? (ii) If f : A ! B is an injection, then how are jAj and jBj related? (iii) If g : C ! D is a surjection, then how are jCj and jDj related? (iv) If h : E ! F is a bijection, then how are jEj and jF j related?
Problem 9: Do Problem 18 on p. 118.
Problem 10: Do Problem 20 on p. 118. Remark: The original problems #11 and #12 on the inclusion-exclusion
principle for 3 sets, have been moved to the 4th HW assignment. 2

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