# HW1 sol - Introduction to Mathematical Analysis SOLUTION...

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Introduction to Mathematical Analysis SOLUTION MANUAL Wieslaw Krawcewicz September 18, 2011 1

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1 Basic Set Theory, Logic and Introduction to Proofs 1.1 Problems 1. Use the Truth table to show that the following statements are tautologies: (a) [( p q ) p ] p (Pierce’s Law); (b) ( p p ) p (Clavius Law); (c) [ p ( q r )] [( p q ) ( p r )] (Law of Conditional Syllogism); (d) ( p q r ) [ p ( q r )]. 2. Check if the following statements are tautologies: (a) ( p q ) [ p ( q r )]; (b) p [( p q ) ( p ∧ ∼ q )]; (c) [( p q ) ( q p )] ( p q ). 3. Check whether the following statements are true or false: (a) “If a is a multiple of 2 and is also a multiple of 7, then if a is not a multiple of 7 implies that a is a multiple of 3 ;” (b) “If it is not true that the line l is parallel to the line m or the line p is not parallel to the line m , then the line l is not parallel to the line m or the line p is parallel to the line m ;” (c) “If James doesn’t know analysis, then if James knows analysis implies that James was born in the 2nd century B.C..” 4. Check if the following quantified statements are true or false: (a) x p ( x ) q ( x ) h x p ( x ) ⇒ ∃ x q ( x ) i ; (b) x p ( x ) ∧ ∃ x q ( x ) ⇒ ∃ x p ( x ) q ( x ) . 5. Prove the following identities for the sets: (a) [ t T ( A t B t ) = [ t T A t [ t T B t ; (b) \ t T ( A t B t ) = \ t T A t \ t T B t ; 2
(c) [ t T ( A t B t ) [ t T A t [ t T B t ; 6. Let A , B , C be sets. Check if the following equalities are true: (a) A ( B C ) = ( A B ) ( A C ); (b) ( A \ B ) C = [( A C ) \ B ] ( B C ); (c) A \ [ B \ C ] = ( A \ B ) ( A C ) 7. Let f : X Y be a function from X into Y . Show that if A and B are subsets of X , then (a) f ( A B ) f ( A ) f ( B ): (b) ( A 0 B 0 ) ( f - 1 ( A 0 ) f - 1 ( B 0 )). (c) f - 1 ( A 0 B 0 ) = f - 1 ( A 0 ) f - 1 ( B 0 ). 8. Show that the function f : X Y is (a) injective if and only if A X f - 1 ( f ( A )) A. (b) surjective if and only if B 0 Y f ( f - 1 ( B 0 )) = B 0 . (c) (b) Show that f is bijective if and only if ( f - 1 ( f ( A )) = A ) ( f ( f - 1 ( B 0 )) = B 0 ) for all A X and B 0 Y . 9. Let f : X Y be a function. Write the logic negation to each of the following statements: (a) f is surjective; (b) f is injective; (c) f is bijective. 3

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1.2 Solutions of Problems 1. Use the Truth table to show that the following statements are tautologies: (a) [( p q ) p ] p (Pierce’s Law); (b) ( p p ) p (Clavius Law); (c) [ p ( q r )] [( p q ) ( p r )] (Law of Conditional Syllogism); (d) ( p q r ) [ p ( q r )]. SOLUTIONS: 1.(a): p q p q ( p q ) p ) [( p q ) p ] p ) 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 The statement [( p q ) p ] p ) is true 1.(b): p p p p ( p p ) p ) 0 1 0 1 1 0 1 1 The statement ( p p ) p is true 1.(c): We denote by ( * ) the statement [ p ( q r )] [( p q ) ( p r )] (for the purpose of fitting it into the truth table). Then we have: p q r q r p q p r p ( q r ) ( p q ) ( p r ) ( * ) 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 The statement [ p ( q r )] [( p q ) ( p r )] is true 1.(d):
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