Exam s 4-4

# Exam s 4-4 - your answer. 3. (16 pts.) Let A be a set and...

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MATH 290 – 10 JUNE 2010 – EXAM 4 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. Deﬁne the relation S on R × R by ( x,y ) S ( z,w ) if and only if x z and y w . (a) (8 pts.) Prove that S is a partial order on R × R . (b) (4 pts.) Is S a linear order ? Why or why not? (c) (4 pts.) Find an upper bound for the rectangle R = [1 , 2] × [3 , 4] = { ( x,y ) R × R :1 x 2 , 3 y 4 } , and ﬁnd sup( R ). You need not verify that your answers are correct. 2. Deﬁne the relation T on R × R by ( x,y ) T ( z,w ) if and only if x + y = z + w . (a) (8 pts.) Prove that T is an equivalence relation. (b) (4 pts.) Describe the set (3 , 4) /T R × R . You can describe the set in words or a picture. You do not have to justify your answer. (c) (4 pts.) Describe the partition of the plane R × R associated to this equivalence relation. You can describe the set in words or a picture. You do not have to justify
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Unformatted text preview: your answer. 3. (16 pts.) Let A be a set and consider the partial order ⊆ on P ( A ). For some natural number n , let B = { B 1 , B 2 , ..., B n } be a family of subsets of A . Prove that the least upper bound (that is, the supremum) of B with respect to the partial order ⊆ is S = n [ k =1 B k . 4. Consider the function f ( x ) = x-1 x + 3 . (a) (4 pts.) Find the implied domain of f . (b) (4 pts.) Find the range of f . (c) (8 pts.) Deﬁne the function g to be the restriction of f to the interval (-3 , ∞ ), that is, g = f | (-3 , ∞ ) . Find Rng ( g ), the range of g . Justify your answer with a proof, that is, prove that for every y ∈ Rng ( g ) there is an x ∈ (-3 , ∞ ) such that f ( x ) = y ....
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## 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.

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