Bridge to Abstract Mathematics: Mathematical Proof and Structures

7.3 EQUIVALENCE CLASSES AND PARTITIONS 243 Additional properties of equivalence classes may be found in Exercise 4. Consider the ...

244 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 -7, -2, 3, 8, 13, 18,.. .I, {... , -11, -6, -1,4,9, 14, 19, ...

7.4 PARTIAL ORDERINGS 245 (g) The equivalence relation - on R defined in Exercise 5(b), Article 7.2 (h) The equivalence relation ...

246 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 EXAMPLE 1 Let S be any set and let A = g(S). Let 5, represe ...

7.4 PARTIAL ORDERINGS 247 of X, we say that L is a least element of X. Analogously, an upper bound U of a subset X of a poset (A ...

248 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 By Theorem 1, a lub or glb, if it exists, is unique. But a ...

7.4 PARTIAL ORDERINGS 249 Two elements x and y of a poset (A, I;) are said to be comparable if and only if either x I y or y I x ...

250 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 (b) Let M and N be subsets of an arbitrary set S. Clearly M ...

7.4 PARTIAL ORDERINGS 251 (a) (i) Calculate the set of all equivalence relations on the set S = (1, 2, 3,4). (ii) Calculate the ...

Relations, Part II: Functions and Mappings CHAPTER 8 At this point you may have some preconceived ideas about functions, re- lat ...

8.1 FUNCTIONS AND MAPPINGS 253 DEFINITION 1 A function is a relation R having the property that if (x, y) E R and (x, z) E y = z ...

W4 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 (2) f(x) = g(x) for all x in the common domain. You are asked to prove this fact ...

8.1 FUNCTIONS AND MAPPINGS 255 Figure 8.1 Graph of the "cubing function." the type described in Exercise 3(b). It should be note ...

256 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 it is. Furthermore, with this understanding in force, we regard two func- tions ...

8.1 FUNCTIONS AND MAPPINGS 257 f (x,) implies x, = x, for all x,, x, E dorn f ." For example, the contra- positive of the defini ...

25(1 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 real numbers, tbese operations are so general as to be applicable to any mappin ...

8.1 FUNCTIONS AND MAPPINGS 259 Figure 8.3 Graph of a function that is its own inverse. Each point (x, y) on the graph of the fun ...

260 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 highly non-one to one and so "do not have inverses" (i.e., their inverse relatio ...

8.1 FUNCTIONS AND MAPPINGS 261 both functions, together with the condition "there exists y.. ." in Defi- nition 5, it clearly ha ...

262 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 In view of Theorem 2, we will dispense henceforth with ordered pair notation in ...