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absolute values additive inverse angle Archimedean ﬁeld Archimedean property assume axioms basic unit vectors brieﬂy called closure law commutativity complete ordered ﬁeld complex numbers consists contains coordinates Corollary countable deﬁned deﬁnition denoted diﬀerent direction cosines disjoint domain dot products E-classes edgelengths equal equation equivalence relation Euclidean space exactly example ﬁeld F ﬁnd ﬁnite ﬁrst formula function value G E1 given Hence Hint holds hyperplane iﬀ implies induction inequalities inﬁnite sequence integers intersection interval isomorphism limxn line segment linear space lower bound mapping f n-tuple natural elements natural numbers norm notation Note obtain one-to-one ordered ﬁeld ordered pairs ordered set particular perpendicular plane points Prove the following rational numbers reader real numbers reﬂexive replaced right-bounded scalar Similarly subset supA supremum triangle inequality trichotomic trichotomy unique unit vectors upper bound vector space Verify write
Page 2 - elements of a set A are also elements of a set B, we say that A is a subset of B, and write A c B. In
Page 4 - (c) The difference A — B is the set of all elements that are in A but not in B
Page 177 - multiply the numerator and the denominator by the conjugate of the denominator to
Page 165 - the points a and b are called the endpoints of the interval.
Page 151 - between a point on one line and a point on the other line.