131aW02-2

# 131aW02-2 - Assignment 2 1 Which of the following...

This preview shows pages 1–2. Sign up to view the full content.

Math 131A/2 Winter 2002 Handout #2 We will have to be comfortable with the notion of an absolute value. We recall that Def. : | x | = ± x if x 0 - x if x 0 ² = ( x 2 ) 1 / 2 You should be able to prove: • ∀ x R , | x | = 0 if and only if x = 0 • ∀ x R , x ≤ | x | • ∀ x,c R , | x | ≤ c if and only if - c x c if and only if ( x c and - x c ) • ∀ x,y R , | x + y | ≤ | x | + | y | • ∀ x,y R , | xy | = | x || y | • || x | - | y || ≤ | x - y | We will use a slightly more general notation for functions than that found in the text. Def. f is a function (from) a set X to a set Y if it is a subset of X × Y such that if ( x,y ) and ( x,y 0 ) f, then y = y 0 . We write f : X ( ) Y and if ( x,y ) f, we also write f ( x ) = y (this is by assumption an unambiguous notation). We let Dom ( f ) = { x X : there exists a y Y with ( x,y ) f } Ran ( f ) = { y Y : there exists an x X with ( x,y ) f } If Dom ( f ) = X we say that f is a function from X to Y, and we write f : X Y (i.e., we omit the parentheses around the arrow).

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Assignment 2 1. Which of the following propositions are true? a) 1 . 3 < √ 3 or chickens have lips. b) 1955 was a bad year for wine and √ 2 can be written as a fraction. c) 6 < 7 or 7 < 8. d) Ducks have claws ⇔ 1 + 1 = 2. e) There are inﬁnitely many primes ⇒ The U. S. debt is \$10. 1 2. § 1.3: 1,2,7,9 3. § 1.4: 5, 6, 8, 9 4. Use the fundamental theorem of arithmetic to prove that 2 1 / 3 is irrational. 5. Assume that x,y, are real numbers. Which of the following propositions are true? a) ( ∀ x )( ∃ y )( x + y = 0) b) ( ∃ y )( ∀ x )( x + y = 0) c) ( ∃ y )( ∀ x )( xy = 0) d) b) ⇒ a) e) a) ⇒ b) 2...
View Full Document

## This note was uploaded on 08/05/2011 for the course MATH 131 taught by Professor Eskin during the Winter '10 term at UCLA.

### Page1 / 2

131aW02-2 - Assignment 2 1 Which of the following...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online