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20. LCI T1 UC u 21. Give a proof using the Pigeonhole Principle: (a) The Italian village of Solomeo has a population of 400. Prove that there are at least two village residents with the same birthday.

20. LCI T1 UC u 21. Give a proof using the Pigeonhole Principle: (a) The Italian village of Solomeo has a population of 400. Prove that there are at least two village residents with the same birthday.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 <n and f: N, → N, thenf is not onto All 18. Let A and B be finite sets with A B. Suppose that f: A → B. (a) If f is one-to-one, prove that f is onto B. ☆ (b) If f is onto B, prove that f is one-to-one. 8x A 92 it ei A Jaz olinnorle (b) wnM gar 19. Prove that if the domain of a function is finite, then the range is finite. 20. Let A be a finite set. Prove that if A N, and A N, then n = m. 21. Give a proof using the Pigeonhole Principle: (a) The Italian village of Solomeo has a population of 400. Prove that there are at least two village residents with the same birthday. 42. Show that the S = 2TRH, wh
Transcribed Image 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 <n and f: N, → N, thenf is not onto All 18. Let A and B be finite sets with A B. Suppose that f: A → B. (a) If f is one-to-one, prove that f is onto B. ☆ (b) If f is onto B, prove that f is one-to-one. 8x A 92 it ei A Jaz olinnorle (b) wnM gar 19. Prove that if the domain of a function is finite, then the range is finite. 20. Let A be a finite set. Prove that if A N, and A N, then n = m. 21. Give a proof using the Pigeonhole Principle: (a) The Italian village of Solomeo has a population of 400. Prove that there are at least two village residents with the same birthday. 42. Show that the S = 2TRH, wh
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
Transcribed Image 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
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