please help me on question 21
Extracted text: I alnd x e N then N, - {x} N._1 (Lemma 5.1.8). 14. Prove Corollary 5.1.10. r-1 Complete the proof of Corollary 5.1.11 by showing that if A is finite and B is a proper subset of A, then B A. osinil ei ( 15. 16. Prove or disprove: (a) If C is an infinite set and C = AU B. then at least one of the sets A or B is infinite. (b) Suppose that A is a set and p is an object not in A. If A AU{p}, then A is infinite. l U2 stinil 17. Prove by induction on n that if r
Nr h - N and h: B N: Then The fU h: AUB Nm which shows that AUB~ Nm+n° AUB is finite. onto onto onto m+n Thus "/>
Extracted text: 70 CHAPTER 5 Cardinality 5.2 {4, 12, 27, 36} and {18, 61} both add up to 79. (c) 2n + 1. (d) Let S be a set of n + 1 integers. Prove that there exist distinct intece and b in S such that a – b is a multiple of n. (e) The English market town of Newton-le-Willows has a population of 21,307. Assuming that every resident has a first, middle, and last name. prove that there are two residents with identical three-letter initials elmom olinil ai norh do *L (f) There are 10 hotel rooms numbered 5, 10, 15, ..., 50. Prove that if 6 nil om b Aid ovom of these rooms are occupied by hotel guests, then at least two occupied rooms have numbers that differ by 5. & bue Aind woda g) There are 50 hotel rooms numbered 1, 2, 3, . . . 50, and 26 of these rooms are occupied by guests. Prove that at least two occupied rooms have numbers that differ by 5. giwollot o l 22. Assign a grade of A (correct), C (partially correct), or F (failure) to eacn ovi oldizzog ofs to Grade orinit i noinu Justify assignments of grades other than A. pinin (a) Claim. If A and B are finite, then A UB is finite. 21 "Proof." If A and B are finite, then there exist m, n E N Such that A N and B N,. Let f: A onitni eh A - N. and h: B > Nr h - N and h: B N: Then The fU h: AUB Nm which shows that AUB~ Nm+n° AUB is finite. onto onto onto m+n Thus