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MAS111_09_quiz2_hints - MAS 111 FOUNDATION OF MATHEMATICS...

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MAS 111 FOUNDATION OF MATHEMATICS QUIZ 2 HINTS 1. Let A = { 1 , 2 , 3 , · · · , 15 } . (25 marks) (a) How many subsets of A contain all of the odd integers in A ? (b) How many 10-element subsets of A contain exactly four odd inte- gers? HINTS : (a) Let E = { 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 } . E is the set of all odd integers in A . The question asks for the number of subsets of A con- taining all odd integers in A , i.e., the number of subsets of A with E as a subset. Let F = A - E = { 2 , 4 , 6 , 8 , 10 , 12 , 14 } . Then any subset of A with E as a subset is of the form E B , where B is a subset of F . As there are 2 7 many such B s, the number of subsets of A containing all integers in A is 2 7 . (b) There are 8 odd integers in A and 7 even integers in B , so the number of 10-element subsets of A containing exactly 4 odd integers (and hence 6 even integers) is ( 8 4 ) · ( 7 6 ) = 490. 2. On the set Z if all integers, define the relation R by (25 marks) R = { ( x, y ) Z × Z : x 2 - y 2 is divisible by 5 } . Prove that R is an equivalence relation. Find the equivalence classes of this equivalence relation on Z . HINTS : It is easy to check that this relation R is reflexive, symmetric and also transitive, and hence it is an equivalence relation.
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  • Spring '10
  • DrChanSongHeng
  • Equivalence relation, Transitive relation, Partition of a set, Preorder, odd integers, FOUNDATION OF MATHEMATICS

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