- MATH 347 SOLUTIONS FOR MIDTERM 1 NOTE 100 = 40 points 1(10 points Suppose C and D are subsets of the domain of a function f(a Prove that f C ∩

Info iconThis preview shows pages 1–2. 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: MATH 347: SOLUTIONS FOR MIDTERM 1 NOTE: 100% = 40 points. 1 (10 points) : Suppose C and D are subsets of the domain of a function f . (a) Prove that f ( C ∩ D ) ⊂ f ( C ) ∩ f ( D ). By the definition of a subset, we have to show that any element of f ( C ∩ D ) is also an element of f ( C ) ∩ f ( D ). To this end, suppose x ∈ f ( C ∩ D ). Then there exists a ∈ C ∩ D s.t. f ( a ) = x . As a ∈ C , we have x = f ( a ) ∈ f ( C ). Similarly, x ∈ f ( D ). Thus, x ∈ f ( C ) ∩ f ( D ). (b) Give an example showing that the inlusion in part (a) may be strict. It may happen that f ( C ∩ D ) is a strict subset of f ( C ) ∩ f ( D ). To provide an example, consider U = { 1 , 2 , 3 } , C = { 1 , 3 } , and D = { 2 , 3 } . Define the function f : U → N by setting f (1) = f (2) = 5 and f (3) = 10. Then f ( C ) = f ( D ) = { 5 , 10 } , hence f ( C ) ∩ f ( D ) = { 5 , 10 } . On the other hand, C ∩ D = { 3 } , hence f ( C ∩ D ) = { f (3) } = { 10 } ....
View Full Document

This note was uploaded on 01/27/2010 for the course MATH 347 taught by Professor ? during the Spring '09 term at University of Illinois at Urbana–Champaign.

Page1 / 2

- MATH 347 SOLUTIONS FOR MIDTERM 1 NOTE 100 = 40 points 1(10 points Suppose C and D are subsets of the domain of a function f(a Prove that f C ∩

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

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