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Of these four types, the complex type declaration may need some de nition; the others should be familiar. The complex type de nition is made up of data that consists of elements, attributes, or element content (whereas the simple type is just character data). The example provided by the W3C is the best so far: < xml version=1.0 > <p:part xmlns:p="http://www.mywebsite.com/PartSchema" xs:schemaLocation = "http://www.mywebsite.com/PartSchema" name="nutbolt"> <mfg>Acme</mfg> <price>10.50</price> </p:part> And the accompanying schema is <xs:schema xmlns:xsd="http://www.w3.org/1999/XMLSchema" targetNamespace="http://www.mywebsite.com/PartSchema"> <xs:element name="part" type="part-type"> <xs:complexType name="part-type"> <xs:element name = "mfg" type="xs:string"/> <xs:element name = "price" type="xs:decimal"/> <xs:attribute name = "name" type="xs:string"/> </xs:complexType> </xs:element> </xs:schema> On further inspection, the preceding code sample can be further reduced to the schema components. Follow the mapping of the components as A1, B1, and so on to get a sense of the underlying construction. This example is from the W3C s XQuery 1.0 and XPath 2.0 Data Model, W3C Working Draft, 7 June 2001. Comments are presented as // A Comment Here // Document node D1 children(D1) = E1



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available for sale sold. The cost of goods sold is then subtracted from Sales to get gross pro t, which is then reduced by operating expenses to determine net income.

Fig. 13-9

(a) Since /C is the largest angle of the triangle, AB is the longest side, or AB OD OF by Principle 12. (b) Since mAC (c) Since OF (d) Since mAB mBC, AC OE, BC BC, the greater arc having the greater chord. mAC by Principle 10. BC; hence,

PROOF:

Reasons Given A midpoint divides a line into two congruent parts. Given A quantity may be substituted for its equal in any inequality. Definition of congruent segments. 5. In a triangle, the larger angle lies opposite the longer side. 6. If unequals are added to unequals, the sums are unequal in the same order.





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We often arrive at a correct conclusion by indirect reasoning. In this form of reasoning, the correct conclusion is reached by eliminating all possible conclusions except one. The remaining possibility must be the correct one. Suppose we are given the years 1492, 1809, and 1960 and are assured that one of these years is the year in which a president of the United States was born. By eliminating 1492 and 1960 as impossibilities, we know by indirect reasoning that 1809 is the correct answer. (Had we known that 1809 was the year in which Lincoln was born, the reasoning would have been direct.) In proving a theorem by indirect reasoning, a possible conclusion may be eliminated if we assume it is true and that assumption results in a contradiction of some given or known fact.

Applying indirect reasoning in life situations Explain how indirect reasoning is used in each of the following situations: (a) A detective determines the murderer of a slain person. (b) A librarian determines which volume of a set of books is in use.

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(a) The detective, using a list of all those who could have been a murderer in the case, eliminates all except one. He or she concludes that the remaining one is the murderer. (b) The librarian finds all the books of the set except one by looking on the shelf and checking the records. He or she concludes that the missing one is the one in use.

Summary

(a) a a (b) a b b b a 88 36 90 2b Ans. Ans. 62 and 26 60 and 30 (c) a a (d) a a b 180 2b 60 b b 180 24 Ans. Ans. 100 and 80 78 and 102

1.1. Point, line, and plane are undefined terms. Which of these is illustrated by (a) the tip of a sharpened pencil; (b) the shaving edge of a blade; (c) a sheet of paper; (d) a side of a box; (e) the crease of a folded paper; (f) the junction of two roads on a map (1.1) 1.2. (a) Name the line segments that intersect at E in Fig. 1-49. (b) Name the line segments that intersect at D. (c) What other line segments can be drawn using points A, B, C, D, E, and F (d) Name the point of intersection of AC and BD. (1.2)

Fig. 1-49

Fig. 1-50

1.3. (a) Find the length of AB in Fig. 1-50 if AD is 8 and D is the midpoint of AB (b) Find the length of AE if AC is 21 and E is the midpoint of AC. 1.4. (a) Find OB in Fig. 1-51 if diameter AD 36.

(1.3)

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