MIT18_014F10_pset1sols

MIT18_014F10_pset1sols - 18.014 Problem Set 1 Solutions...

Info iconThis preview shows pages 1–3. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.014 Problem Set 1 Solutions Total: 24 points Problem 1: If ab = 0, then a = 0 or b = 0. Solution (4 points) Suppose ab = and b = 0. By axiom 6, there exists a real number y such that by = 1. Hence, we have a = 1 a = a 1 = a ( by ) = ( ab ) y = 0 y = 0 using axiom 4, axiom 1, axiom 2, and Thm. I.6. We conclude that a and b cannot both be non-zero; thus, a = 0 or b = 0. Problem 2: If a < c and b < d , then a + b < c + d . Solution (4 points) By Theorem I.18, a + b < c + b and b + c < d + c . By the commutative axiom for addition, we know that c + b = b + c,d + c = c + d . Therefore, a + b < c + b,c + b < c + d . By Theorem I.17, a + b < c + d . Problem 3: For all real numbers x and y , | x | | y | | x y | . Solution (4 points) By part (i) of this exercise, | x || y | | x y | . Now notice that ( | x || y | ) = | y || x | . By definition of the absolute value, either || x || y || = | x || y | or || x || y || = | y || x | . In the first case, by part (i) of this problem, we see that || x | | y || | x y | . In the second case, we can interchange the x and y from part (i) to get || x = | | y || | y | | x | | y x | = | x y | , where the last equality comes from part (c) of this problem. Thus, || x | | y || | x y | . Problem 4: Let P be the set of positive integers. If n,m P , then nm P . Solution (4 points) 1 Fix n P . We show by induction on m that nm P for all m P ....
View Full Document

This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

Page1 / 6

MIT18_014F10_pset1sols - 18.014 Problem Set 1 Solutions...

This preview shows document pages 1 - 3. Sign up to view the full document.

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