Examples of Categories

The best way to get acquainted with categories is to look at the numerous examples that can be found in almost every branch of mathematics. 

We note that A and B are isomorphic in Sets if and only if they have the same cardinality. 

We note that G and H are isomorphic in Grp if and only if the groups G and H are isomorphic as groups. 

We note that H and G are isomorphic in Ab if and only if the abehan groups 

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The currently accepted mechanism is essentially that proposed by Westheimer. The first two steps constitute an example of category 4 (p. 1508). 

An example of category (b) is the four-site skeleton of Fig. 17, with symmetry C4 . Here the improper operations (24) and (13) determine the ( —1) = 3-dimensional hypersurfaces... 

Examples of Category 3 applications include changes to the specifications of the active ingredient, change of shelf life or storage conditions, and change of trade name. 

An example of category 1 is found in the observation by Brown and Liu that eliminations from the rigid ring system 44, induced by the sodium salt of 2-cyclohexylcyclohexanol in triglyme, produces norborene (98 percent) but no 2-deuteronorbornene.103 The dihedral angle between D and tosylate is 0°, but... 

Figure 15-1. Examples of categories of figure art (A) line art, (B) halftone or continuous-tone art, (C) grayscale line art, and (D) a combo.

Table 10.1 Example of categories used in an inspection report (source Soil Association Certification Ltd processing and packing inspection report used with permission).

Examples of category 1 were established first with the sesquiterpene... 

In our catalog, we have considered three types of components with hydroxyl groups (1) components with a carboxyl group and its hydroxyl group (discussed in Chapter IV), (2) components with an hydroxyl group attached to a monocyclic or polycyclic benzenoid nucleus, that is, a phenol (discussed in Chapter IX), and (3) an hydroxyl group attached to a saturated or unsaturated aliphatic, alicyclic, or nonbenzenoid nucleus, which may or may not include another functional entity. An example of category (3) is the first item in Table... 

Giving examples of categories Eight common categories like CLOTHING with a large number of concrete instances were used. Normative data were available on... 

Examples of categories in this dimension include the following (the list is not exhaustive) ... 

An example of category I is the preparation of optically active epibromohydrin by base-catalyzed hydrolysis and dehydrogenation of the monoester of phthalic acid, as shown in reaction 18.15. 

Ester formation is a typical example of category 2. Although other methods such as the use of distillation column reactors (see Chapter 25) are preferred, layer separation can also be used by adding dichloromethane to remove the water. A well-known example of biphasing in organic synthesis belongs to this class, the Schotten-Baumann acylation reaction... 

In the following sections, we will discuss the MCMP methods—goal programming, cempromise programming, and interactive methods, as examples of categories 1,2, and 3 type approaches. 

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