Homework due on Monday, Jan. 24:
1.5:3dg,5c plus show that sqrt{3} is not a rational number; here, "sqrt" stands for "square root".
Homework due on Monday, Jan. 31:
2.1: 1ab,2ab,8,9,14b
Plus, use dual inclusion to prove Theorem 2.2.1 parts g and n, and Theorem 2.2.2 part g. (These are the 3 formulas that were assigned in lecture.)
Plus, prove that if X and Y are sets, then X \subset Y if and only if X \cup Y =Y, where "\cup" stands for "union". Here "if and only if" means that you have two things to prove, namely, you have to show that if X \cup Y =Y, then X \subset Y, and you also have to show that if X \subset Y, then X \cup Y =Y.
Plus, if A and B are sets, use "set algebra" to simplify A-(A-B). Be sure to justify your steps by quoting a rule that applies.
Our homework grader has now been assigned, so please have your homework ready to hand-in on Monday, Jan. 31 at lecture.
Homework due Monday, Feb.7:
1.3: 1abf, 2abf, 8abgh, 10abeg
Use the definition of the limit of a sequence to prove that lim_{n \to \infty} (3n-10)/n = 3; and prove that lim_{n \to \infty} (3n-10)/n is not 7/2 by using the negation of the definition.
Homework due Monday, Feb. 13:
(1) Verify the \epsilon, N definition that lim_{n \to \infty} (3n+75)/(n^4+10) = 0.
(2) Give the negation of lim_{n \to \infty} a_n = + \infty by using a properly quantified English sentence.
(3) Use the M, N definition to prove that lim_{n \to \infty} (n^2 -100n)^{1/3} = +\infty.
(4) Use the appropriate definition to verify that lim_{n \to \infty} (1 -sqrt{2n}) = - \infty. Remember that sqrt means "square root".
OUR FIRST EXAM WILL BE HELD ON THURSDAY, MARCH 3, DURING THE LECTURE PERIOD
Homework due Monday, Feb, 20:
(1) Use versions of the triangle inequality to show that
(a) 4sin2x -3x is bounded for all x with |x| <4;
(b) 1/(4sin2x -3x) is bounded for all x with |x| >2.
(2) Prove the first part of the Simple Limit Theorem, namely
If {a_n} is a sequence of real numbers, c and L are real numbers, and lim_{n \to \infty} a_n = L, then lim_{n \to \infty} ca_n = cL. (Hint: Consider the case c=0 separately.)
Homework due Monday, Feb. 27:
2.4: 6el, 7ck, 8e,9a
Homework due Monday, March 7:
(1) Disprove: If {a_n} and A satisfy lim |a_n| = |A|, then lim a_n = A.
(2) Use the squeeze theorem to show that lim_{n \to \infty} (n!)/(n^n) =0.
(3) Give examples of sequences {a_n} and {b_n} with limits +\infty such that the sequence {a_n-b_n} has limit 0, has limit 1, has limit +\infty, and has limit -\infty. Of course, there should be different examples for each conclusion. For the first one, give an example where for each n, a_n and b_n are different
Homework due Monday, March 21:
2.2: 15cd, 16a. For 16a, we proved in class that the set on the left side is always contained in the set on the right side. To show that the two sides are not always equal, you need to give an example!
Plus: If A X B = A X C and A is not the empty set, show that B=C. Be sure that you indicate how you use the fact that A isn't empty.
3.2: 1dhi, 2bc, 5aeh,11,12
Homework due Monday, March 28:
3.3: 2abc, 3ad, 7
1.7: 9abc (This refers to the Division Algorithm.)
Homework due Monday, April 4:
1.7: 11,13ac,14cd,15,16b,18 with the following correction: replace the words "an integer solution" by "integer solutions x,y".
Plus Prove or Disprove:
(a) The operation x o y = x/y of ordinary division is a binary operation on the set of natural numbers.
(b) For any two 2x2 matrices A and B, the operation A o B = AB of matrix multiplication is a binary operation on the collection of all 2 x 2 matrices.
EXAM 2 WILL BE ON APRIL 18.
Homework due Monday, April 11:
6.1: 1bgh, 2 for bgh, 3 for bgh, 5 for acd
Plus:
(1) Prove part (b) of Theorem 6.1.3, i.e., prove that the operation of addition mod m on Z_m defined by [x] +_m [y] = [x+y] is associative, has identity [0], and that [i] has inverse [-i] for every i in Z.
(2) Make Cayley tables for Z_8 for both mutiplication mod 8 and addition mod 8. Identify all elements of Z_8 which have inverses with respect to the operation of multiplication mod 8.
Homework due THURSDAY, April 21 (Note change of day.)
(1) If (G,#) is a group and if a#a = e (the identity) for all a in G, show that (G,#) is Abelian.
6.2: 8c,17
Homework due Monday, April 25:
For n in N, let S_n be the collection of all permutations of n objects. You may call the objects 1,2,...,n. Use induction to show that S_n has exactly n! distinct elements.
PLUS 6.5: 5
Homework for Monday, May 2:
(1) Explain why (Z_6, ._6) is not a group. Is (Z_6,+_6,._6) a field?
(2) Find the inverse of the complex number 3-4i. Verify that your answer is correct by showing that its product with 3-4i equals 1.
(3) Express the complex number (1-i)/(2+3i) in the form a+ib for appropriate real numbers a and b.
(4) For the complex numbers, verify the field property z_1z_2 = z_2z_1 of commutativity of multiplication of complex numbers z_1 and z_2. Also verify the distributive property z_1(z_2+z_3) = z_1z_2 + z_1z_3 for complex numbers.
OUR FINAL EXAM WILL BE ON WEDNESDAY, MAY 11, 12:00-3:00 PM, IN SCOTT 121, OUR USUAL LECTURE ROOM
WE WILL HAVE A REVIEW SESSION ON WEDNESDAY, MAY 4, 2:00-4:00 PM, IN SCOTT 121.
SOME REVIEW PROBLEMS
(1) Let S be a set of real numbers. A real number x is said to be an interior point of S is there exists \delta >0 such that the interval (x-\delta, x+\delta) is contained in S. The interior of S, denoted Int(S), is defined to be the set of all interior points of S.
(a) State precisely what it means to say that a real number x is not an interior point of S.
(b) Show that 0 is not an interior point of the interval [0,1].
(c) If S_1 and S_2 are two sets of real numbers, prove that Int(S_1) U Int(S_2) is a subset of Int(S_1 U S_2), where "U" denotes union.
(2) Use induction to prove that for all natural numbers n, 5^n-4n-1 is divisible by 16.
(3) Use the logical negation of the definition of limit to prove that lim_{x \to 0} x/(x+1) is not equal to 1.
(4) Use case analysis to show that for any two real numbers x and y, it is true that |xy| = |x||y|.
(5) Let S be a set of real numbers. A real number x_0 is said to be a maximum of S (respectively, minimum of S) if both (1) x_0 belongs to S, and (2) x_0 >= x for every x in S (respectively, x_0 <= x for every x in S). Also, if such an x_0 exists, we say that S has a maximum (respectively, a minimum).
(a) Show that if S has a maximum x_0, then x_0 is unique.
(b) Without using the words 'no', 'not', etc., explain what it means to say that a real number x_0 is not the maximum of S. Explain what it means to say that S does not have a maximum.
(c) Show that the interval (0,1) does not have a maximum.
(d) Define -S = {x: -x \in S}. Show that if S has a maximum, then -S has a minimum, and that minimum(-S)= -maximum(S).
(6) Use the definition of limit of a sequence to show that lim_{n \to \infty} (n^2+1)/(3n^3-1) =0
OUR GRADER KEVIN CALKINS WILL HOLD AN ADDITIONAL REVIEW SESSION ON SUNDAY, MAY 8, 7:00-9:00 PM IN OUR REGULAR ROOM SCOTT 121. ALSO, HE WILL BE AVAILABLE TO ANSWER QUESTIONS AT KREEGER LEARNING CENTER ON MAY 3 FROM 11:00-4:00; SUNDAY MAY 8 FROM 2:00-4:00; MONDAY, MAY 9 FROM 11:00-4:00.