homework1solutions_Math 113

# homework1solutions_Math 113 - Math 113 Section 5 Homework#1...

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

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: Math 113 Section 5 Homework #1 Solutions Fall 2010 Due: Wednesday, September 8 1. Section 0.1, Exercise 5: Determine whether the following functions f are well defined: (a) f : Q → Z defined by f ( a b ) = a . f is not well defined: Let a b be another representative of a b (that is, a Ó = a and b Ó = b but a b = ab ). Then f ( a b ) = a Ó = a = f ( a b ) . (b) f : Q → Q defined by f ( a b ) = a 2 b 2 . f is well defined: This is because the codomain is Q . Let a b be another representative of a b . Then a b = ab = ⇒ ( a b ) 2 = ( ab ) 2 = ⇒ a 2 b 2 = a 2 b 2 = ⇒ a 1 b 2 ∼ a b = ⇒ f ( a b ) = f ( a b ) . 2. Prove Proposition 2, section 0.1: Let A be a nonempty set. (a) Prove that if ∼ determines an equivalence relation on A then the set of equivalence classes A/ ∼ of ∼ forms a partition of A . Let a ∈ A and ¯ a = { b ∈ A | b ∼ a } . We need to show (1) t a ∈ A ¯ a = A and (2) ¯ a ∩ ¯ b = φ if ¯ a Ó = ¯ b . (1) Every element is in some equivalence class. In particular, a ∈ ¯ a . (2) Assume ¯ a ∩ ¯ b Ó = φ . Then there exists a c ∈ ¯ a ∩ ¯ b . Therefore c ∼ a and c ∼ b . Because ∼ is an equivalence relation, a ∼ b so a ∈ ¯ b and b ∈ ¯ a . Further, for any c ∈ ¯ a , c ∈ ¯ b and vice versa. Thereforeand vice versa....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

homework1solutions_Math 113 - Math 113 Section 5 Homework#1...

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

View Full Document
Ask a homework question - tutors are online