construction some qus

Construction some - MATH 30A FALL 04 Homework This is homework from Math 30a Fall 2004 I was teaching only one course So I had a lot of time to

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 30A FALL 04 Homework This is homework from Math 30a, Fall 2004. I was teaching only one course. So I had a lot of time to prepare the course. • The book was John Fraleigh ”A first course in Abstract Alge- bra,” 7th edition. We covered the first four chapters. • About 3/4 of the course was planned to be on group theory (Chapters 1,2,3). The last quarter (Chapter 4) covers basic concepts of rings and fields. • There were several practice quizzes, a midterm and a take-home final. Date : August 30, 2006. 1 1. Homework 1 State the questions and answer them in complete sentences or as you would speak. (Symbols are nouns and equations are also sentences. E.g, ” a = b but c 6 = d ” is a sentence.) 12) Decide whether these relations are (1) functions, (2) 1-1, (3) onto. a) is a function. It is not 1-1 since f ( a ) = f (2). It is not onto since 2 is not in the range. b) is a function but it is not 1-1 and not onto. c) is not a function since 1 maps to more than one element and 2,3 don’t map to anything. d) is a bijection (a function which is 1-1 and onto). e) is a function which is neither 1-1 nor onto. f) is not a function since 2 ∈ A maps to more than one element of B . 15 Show that the open interval S = (0 , 1) has the same cardinality as R . Proof. We will show that the function f : (0 , 1) → R is given by f ( x ) = tan(( x- 1 / 2) π ) is a bijection. ( function ) tan y ∈ R is defined for all real y so f is a function. ( 1-1 ) The derivative of f ( x ) is , f ( x ) = πsec 2 ( x- 1 / 2) π = π cos 2 ( x- 1 / 2) π . This is finite and positive as long as the denominator is nonzero. But cos( x- 1 / 2) π = 0 only if an odd multiple of π/ 2, i.e., when x is an integer. So f is 1-1. (The rule is that a differentiable function on an interval is 1-1 if (but not only if) its derivative is always positive on the interval.) ( onto ) Take any y ∈ R . Then- π/ 2 < tan- 1 y < π/ 2. So, < tan- 1 y + π/ 2 π < 1 If we let x = tan- 1 y + π/ 2 π then π ( x- 1 / 2) = πx- π/ 2 = tan- 1 y and y = f ( x ). So, f is onto. 16 (a) The empty set has one subset (itself). In general a set with n elements has 2 n subsets. This was explained in class. 1 2. Homework 2 9) Calculate (1- i ) 5 using the binomial expansion. (1- i ) 5 = 1- 5 i + 10 i 2- 10 i 3 + 5 i 4- i 5 = 1- 5 i- 10 + 10 i + 5- i =- 4 + 4 i 21 Find all solutions of the equation z 6 =- 64. One solution is z = 2 e πi/ 6 = √ 3+ i . There are three other conjugates of this number given by √ 3- i,- √ 3 + i,- √ 3- i And there are two more solutions: ± 2 i . There are no other solutions since the ratio of any two solutions is a 6th root of unity: z z 6 = z 6 z 6 =- 64- 64 = 1 ....
View Full Document

This note was uploaded on 03/24/2012 for the course MATH 203 taught by Professor Ankit during the Spring '12 term at Evergreen.

Page1 / 39

Construction some - MATH 30A FALL 04 Homework This is homework from Math 30a Fall 2004 I was teaching only one course So I had a lot of time to

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online