811Solutions8

# 811Solutions8 - Homework 7 Solutions Kyle Chapman 153.4 We...

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: Homework 7 Solutions Kyle Chapman November 30, 2011 153.4 We seek to show that, given a partion of S into the collection S i , there is a unique equivalence relation with the property that each S i forms an equivalence class. First we will show existence. Let ∼ be defined by a ∼ b if there exists some i such that both a and b lie in S i . For reflexsivity, we know that the collection of S i cover S so for every a in S there exists some i such that a lies in S i so a ∼ a . For symmetry, it is clear that if a ∼ b then there is some i so that both a and b lie in S i so b ∼ a . Finally, for transitivity, suppose that a ∼ b and b ∼ c . Then there is some i such that a and b both lie in S i and some j so that b and c both lie in S j . Since b lies in both S i and S j , they must be equal, so a and c both lie in S i . Thus, a ∼ c . We now know that ∼ is an equivalence relation. Next we show that it’s equivalence classes are the collection of S i . We know that the equivalence classes form a partition, so it suffices to show that if an....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

811Solutions8 - Homework 7 Solutions Kyle Chapman 153.4 We...

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

View Full Document
Ask a homework question - tutors are online