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
Unformatted text preview: MATH 290 EXAM 4 SOLUTIONS 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. Solution: (a). First we need to show that S is reflexive. Let ( x,y ) R R . Since x x and y y , ( x,y ) S ( x,y ) and S is reflexive. Now we will show that S is antisymmetric. Let ( x,y ) , ( z,w ) R R . Assume that ( x,y ) S ( z,w ) and that ( z,w ) S ( x,y ). We must show that ( x,y ) = ( z,w ), that is, that x = z and y = w . Since ( x,y ) S ( z,w ), x z and since ( z,w ) S ( x,y ), z x . Hence x = z . Also, since ( x,y ) S ( z,w ), y w and since ( z,w ) S ( x,y ), w y . Hence y = w ....
View
Full
Document
 Fall '08
 Alligood,K
 Math, Advanced Math

Click to edit the document details