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Computer Arithmetic and Validity
book

Computer Arithmetic and Validity

by Ulrich Kulisch
April 2013
Intermediate to advanced content levelIntermediate to advanced
456 pages
16h 7m
English
De Gruyter
Content preview from Computer Arithmetic and Validity
24 Chapter 1 First concepts
(a) If fM , g is a lattice, then fS, g is a sublattice.
(b) If fM , g is a complete lattice and
S :D S [fi.M/g,thenfS, g is a complete
sublattice.
Proof. (a) Let x, y 2 S.Thenx a and y a and therefore inf
M
fx, yg
sup
M
fx, yga, i.e., inf
M
fx, yg,sup
M
fx, yg2S, which proves the assertion
by Corollary 1.16.
(b) Let ¿ ¤ A S. Then for all x 2 S, x a, and therefore inf
M
S sup
M
S
a, i.e., inf
M
S,sup
M
S 2 S .But¿ S also. Then in order to apply Corol-
lary 1.16, inf
M
¿ D i.M/ and sup
M
¿ D o.M/ must be elements of S.Since
o.M/ a, it is automatically an element of S and therefore of
S .Thati.M/ 2 S,
however, has been assumed ...
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Publisher Resources

ISBN: 9783110301731