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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 yy 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.
 Fall '08
 STAFF
 Math, Calculus

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