Bridge to Abstract Mathematics: Mathematical Proof and Structures

8.1 FUNCTIONS AND MAPPINGS 263 We conclude this article with a rather abstract result that relates the con- cept of one to one t ...

284 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 (a) Prove that two functions f and g are equal (i.e., contain precisely the sam ...

8.1 FUNCTIONS AND MAPPINGS 265 (c) Prove that iff and g are linear mappings of R into R, that is f(x) = Mx + B and g(x) = Nx + C ...

286 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 8.2 More on Functions and Mappings- Surjections, Bijections, Image, and Inverse ...

8.2 MORE ON FUNCTIONS AND MAPPINGS 267 assumption, then (g 0 f )(x) = (h 0 f )(x) so that g( f (x)) = h( f (x)). Since y = f(x), ...

268 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 Each of the remaining results (a), (b), and (c) can be proved in a similar direc ...

8.2 MORE ON FUNCTIONS AND MAPPINGS 269 are both one to one and onto. The latter combination is of sufficient im- portance to war ...

270 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 for Article 8.3, let us take an informal look at possibilities for injections, s ...

8.2 MORE ON FUNCTIONS AND MAPPINGS 271 (c) A = f-'(B) (d) rng f = f(A) (e) If MI, M, c A with M, c M,, then f(M,) G f(M2) (f) If ...

272 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 f - '( f (M)) c M. Let x E f - '( f (M)). Then f (x) E f (M) so that f (x) = f(m ...

8.2 MORE ON FUNCTIONS AND MAPPINGS 273 Exercises Use Definition 1 and the definition of rng f (Definition 4, Article 7.1) to pr ...

274 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 (c) Use the result of Theorem 4, Article 8.1, to give a proof of (a) of Theorem ...

8.2 MORE ON FUNCTIONS AND MAPPINGS 275 (6) Prove (b) of Theorem 6; that is, iff: A -, B, then f (MI u M,) = f (MI) u f (M,) for ...

276 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 (b) Prove that iff is one to one, then Sf is one to one [recall Exercise 8(b)(ii ...

8.3 CARDINAL NUMBER OF A SET 277 (d) Suppose a E A - im g or b E B - im f. What is the appropriate conclusion about ancestry in ...

278 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 Justification for the name "numerical equivalence" is provided by the following ...

8.3 CARDINAL NUMBER OF A SET 279 EXAMPLE 3 Prove that the set N of all positive integers is in one-to-one correspondence with 2N ...

280 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 positive integer, starting with 1 and ending with some n. We then designate n by ...

8.3 CARDINAL NUMBER OF A SET 281 N, they have just as many elements as N! In fact, it can be proved that N is numerically equiva ...

282 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 0 f(i) -f(i) f(2) -f(2) fi3) -f(3) f(4) Figure 8.7 Write down an explicit rule d ...