# HW2 sol - Introduction to Mathematical Analysis SOLUTION...

This preview shows pages 1–5. Sign up to view the full content.

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

View Full Document

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: Introduction to Mathematical Analysis SOLUTION MANUAL Wieslaw Krawcewicz October 5, 2011 1 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 ) ) ⇒ [ ∃ x p ( x ) ⇒ ∃ x q ( x ) ] ; (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 ′ ⊂ B ′ ) ⇒ ( f − 1 ( A ′ ) ⊂ f − 1 ( B ′ )). (c) f − 1 ( A ′ ∩ B ′ ) = f − 1 ( A ′ ) ∩ f − 1 ( B ′ ). 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 ′ ⊂ Y f ( f − 1 ( B ′ )) = B ′ . (c) (b) Show that f is bijective if and only if ( f − 1 ( f ( A )) = A ) ∧ ( f ( f − 1 ( B ′ )) = B ′ ) for all A ⊂ X and B ′ ⊂ 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 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 ⇒...
View Full Document

## This note was uploaded on 02/23/2012 for the course MATH 4301 taught by Professor Krwaw during the Fall '11 term at King Mongkut's Institute of Technology Ladkrabang.

### Page1 / 33

HW2 sol - Introduction to Mathematical Analysis SOLUTION...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online