1873_solutions

In p given any number x p we have log x x 1 q and so

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Unformatted text preview: ws at once from Exercise 11. 14. Suppose that S is a linearly independent subset of a vector space V. Prove that the relation r defined by V V S Þ x y is linearly dependent r  x, y is an equivalence relation in V. Given x S we know from the fact that the set S Þ x x contains the additive identity 0 that S Þ x x is linearly dependent. Therefore the relation r is reflexive. Given x and y in S, the set S Þ x y is linearly dependent if and only if the set S Þ y x is linearly dependent. Therefore r is symmetric. Finally suppose that x, y and z belong to S and that x r y and that y r z. The sets S Þ x y and S Þ y z are linearly dependent. Since both x y and y z must lie in the span S è of S we know 46 yy that x S è and so the set S Þ x z z is linearly dependent. 15. What can be said in the above exercise if the set S is linearly dependent? If S is linearly dependent then the condition x r y holds for all x and y in S, in other words, r  S which is an equivalence relation in S. S Exercises on Order Relations 1. Suppose that  is the relation in R 2 defined by a, b  x, y if and only if a x and b y, where is the usual order in R. Prove that  is a partial order in R 2 but is not a total order. It is clear that the relation  is a partial order. Since neither of the points 0, 1 and 1, 0 precedes the other. 2. Suppose that  is the relation in R 2 defined by a, b  x, y if and only if a usual order in R. Prove that  is a partial order in R 2 but is not a total order. This exercise is just like Exercise 1. x and b 3. Suppose that  is the relation in R 2 defined by a, b  x, y if and only if either a the usual order in R. Is the relation  a partial order in R 2 ? This relation fails to be transitive. Note that 0, 4  1, 0  1, 2 . y, where x or b is the y, where is 4. Suppose that  is the relation in R 2 defined by a, b  x, y if and only if either a  x or a  x and b y . Prove that  is a total order in R 2 . We see at once that the relation  is reflexive and it is also clear that if a, b  x, y and a, b  x, y then a  x and b  y. To see that  is transitive, suppose that a, b  x, y  u, v . If either a  x or x  u then we have a  u and the condition a, b  u, v is assured. Otherwise a  x  u and it follows from the fact that b y v that b v. So in this case we again have a, b  u, v . Finally suppose that a, b and x, y are any points in R 2 . If a  x then we have a, b  x, y and if x  a then we have x, y  a, b . In the event that a  x then the condition a, b  x, y holds when b y and the condition x, y  a, b holds when y b. We conclude that  is a total order in R 2 . 5. Suppose that is a family of sets and that r is the relation in that consists of all pairs A, B such that either A  B or A B . Is r always a partial order? This relation fails to be transitive. Look at the sets 1 and 2 and 1, 3 . 6. Given a set S, is S S a partial order of S? No it isn’t. If x and y are any two different members of S then, although both x, y and...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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