Exam s 3-4

# Exam s 3-4 - ˆ n k =1 A k = n k =1 B ∩ A k 4(a(12 pts...

This preview shows page 1. Sign up to view the full content.

MATH 290 – 3 JUNE 2010 – EXAM 3 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This test will be counted out of a total of 60 points. 1. (16 pts.) Prove by induction that for all natural numbers n , n X k =1 ( k · k !) = ( n + 1)! - 1. 2. (16 pts.) Prove by induction that for all natural numbers n 4, 2 n < n !. 3. (16 pts.) Suppose that { A n : n N } is an indexed family of sets indexed by the natural numbers, and let B be a set. Prove by induction that for all natural numbers n , B
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ˆ n [ k =1 A k ! = n [ k =1 ( B ∩ A k ) . 4. (a) (12 pts.) For all sets A , B , C , D , prove that ( A × B ) ∪ ( C × D ) ⊆ ( A ∪ C ) × ( B ∪ D ) . (b) (4 pts.) Find an example of sets A , B , C , and D where the inclusion in part (a) is proper, that is, where ( A × B ) ∪ ( C × D ) 6 = ( A ∪ C ) × ( B ∪ D ). You do not have to provide a proof that your example works, a picture or similar justiﬁcation will be suﬃcient. (Hint: Try intervals on the real line.)...
View Full Document

## This note was uploaded on 03/05/2011 for the course MATH 290 taught by Professor Alligood,k during the Fall '08 term at George Mason.

Ask a homework question - tutors are online