131aW02-2

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

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

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