{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lec-37 - Administration 1 Midterm regrades due today 2...

This preview shows pages 1–17. 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 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 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 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: Administration 1. Midterm regrades due today! 2. Homework due on Friday. CS70: Satish Rao: Lecture 37. I Countability. I Uncountability. Cardinality. A bijection is function f : S → T , Cardinality. A bijection is function f : S → T , I one-to-one, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) Cardinality. A bijection is function f : S → T , I one-to-one, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Cardinality. A bijection is function f : S → T , I one-to-one, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Cardinality. A bijection is function f : S → T , I one-to-one, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Two sets, S and T have the same cardinality if there is a bijection from S to T Cardinality. A bijection is function f : S → T , I one-to-one, ∀ x , y ∈ S , x 6 = y = ⇒ f ( x ) 6 = f ( y ) I and onto, ∀ y ∈ T , ∃ x , y = f ( x ) . Two sets, S and T have the same cardinality if there is a bijection from S to T If there is a bijection from S to T , there is one from T to S Countable. Countable. Definition: S is countable if there is a bijection between S and some subset of N . Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Bijection to or from natural numbers implies countably infinite. Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Bijection to or from natural numbers implies countably infinite. Enumerable means countable. Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality . If the subset of N is infinite, S is countably infinite . Bijection to or from natural numbers implies countably infinite. Enumerable means countable. Subset of countable set is countable. Countable. Definition: S is countable if there is a bijection between S and some subset of N . If the subset of N is finite, S has finite cardinality ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 169

lec-37 - Administration 1 Midterm regrades due today 2...

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online