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Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 21E
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1. Let X be a set, and suppose that f: Z, → X is an injection. Prove that X is infinite. 2. Suppose that AC {1,2,..., 12} with |A| > 5. Show that there exist a, b, c, d e A such that (a, b) + (c, d) and a +b = c + d. 3. Suppose that A C {1,2,..., 100} with |A| > 10. Show that there exist nonempty subsets B, C C A such that B,C are disjoint and the sum of the elements of B is equal to the sum of the elements of C. 4. Fix n, k E Z, with k < n, and let A1 Determine (with proof) the cardinality of the set {1,2, ..., k}, A2 {k + 1, k + 2, ..., n}. {f : N, N„|f(A)) C A1, ƒ (A2) C (A2)}. 5. (optional) Use induction to prove the following generalized version of the inclusion- exclusion principle: If n E Z4 and A1,..., An are finite sets, then E(-1)"I-1 i=1 07ICN, where Eo4ICN, means to take a sum as I ranges over all nonempty subsets of N,, and for I = {i1,..., im} C {1,...,n}, N A; is defined to be iɛl NA: = A;, N A, n..n m. ieI Hint: at some point you will need to use the fact that 'n-1 n-1 U A.)n A, = U(A;N A.). i=1 i=1 6. (a) Fix n, m e Z4. For each i e N,, define A; = {f : Nm → N„|i ¢ im(f)}. Prove (by constructing a bijection) that for any I C N,, with |I| = k < n, (n – k)". iel (b) Use problem 5 and part (a) to prove that for n, m e Z4, the number of surjections from Nm to N,, is given by (п — 1)" + n-1 (n – 2)m – ...+ (-1)" n – 1
Transcribed Image Text:1. Let X be a set, and suppose that f: Z, → X is an injection. Prove that X is infinite. 2. Suppose that AC {1,2,..., 12} with |A| > 5. Show that there exist a, b, c, d e A such that (a, b) + (c, d) and a +b = c + d. 3. Suppose that A C {1,2,..., 100} with |A| > 10. Show that there exist nonempty subsets B, C C A such that B,C are disjoint and the sum of the elements of B is equal to the sum of the elements of C. 4. Fix n, k E Z, with k < n, and let A1 Determine (with proof) the cardinality of the set {1,2, ..., k}, A2 {k + 1, k + 2, ..., n}. {f : N, N„|f(A)) C A1, ƒ (A2) C (A2)}. 5. (optional) Use induction to prove the following generalized version of the inclusion- exclusion principle: If n E Z4 and A1,..., An are finite sets, then E(-1)"I-1 i=1 07ICN, where Eo4ICN, means to take a sum as I ranges over all nonempty subsets of N,, and for I = {i1,..., im} C {1,...,n}, N A; is defined to be iɛl NA: = A;, N A, n..n m. ieI Hint: at some point you will need to use the fact that 'n-1 n-1 U A.)n A, = U(A;N A.). i=1 i=1 6. (a) Fix n, m e Z4. For each i e N,, define A; = {f : Nm → N„|i ¢ im(f)}. Prove (by constructing a bijection) that for any I C N,, with |I| = k < n, (n – k)". iel (b) Use problem 5 and part (a) to prove that for n, m e Z4, the number of surjections from Nm to N,, is given by (п — 1)" + n-1 (n – 2)m – ...+ (-1)" n – 1
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