klarfeld-sec-3-3-3-5 - Jennifer Klarfeld Homework 9 Section...

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Jennifer Klarfeld – Homework 9 – Section 3.3/3.5 – 8 th Edition Section 3.3 2. For the given set A , determine whether P is a partition of A . (a) A ={ 1,2,3,4 } ,P ={{ 1,2 } , { 2,3 } , { 3,4 }} i. For all sets in P , the set is nonempty. ii. Let X ={ 1,2 } and Y ={ 2,3 } . Since X ≠Y and X ∩Y ≠ , and since one of those conditions must be true for P to be a partition, P is not a partition of A . 2. (continued) 1
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Jennifer Klarfeld – Homework 9 – Section 3.3/3.5 – 8 th Edition (b) A ={ 1,2,3,4,5,6,7 } , P ={{ 1,2 } , { 3 } , { 4,5 }} i. For all sets in P , the set is nonempty. ii. There are no two sets equal to each other and no two sets have a common element, so for all X and Y in P , X ∩Y = . iii. ¿ X P X = { 1,2 } { 3 } { 4,5 } = { 1,2,3,4,5 } ≠ A. So P is not a partition of A . 2. (continued) 2
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Jennifer Klarfeld – Homework 9 – Section 3.3/3.5 – 8 th Edition (c) A = { 1,2,3,4,5,6,7 } ,P = { { 1,3 } , { 5,6 } , { 2,4 } , { 7 } } i. For all sets in P , the set is nonempty. ii. There are no two sets equal to each other and no two sets have a common element, so for all X and Y in P , X ∩Y = . (b) ¿ X P X = { 1,3 } { 5,6 } { 2,4 } { 7 } = { 1,,2,3,4,5,6,7 } = A. Therefore, P is a partition of A . 3
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Jennifer Klarfeld – Homework 9 – Section 3.3/3.5 – 8 th Edition 9. List the ordered pairs in the equivalence relation on A={1, 2, 3, 4, 5} associated with these partitions: (b) {{1}, {2}, {3,4}, {5}} ´ 1 ={ 1 } , ´ 2 ={ 2 } , ´ 3 = ´ 4 ={ 3,4 } , ´ 5 ={ 5 } The ordered pairs in the equivalence relation on A are: {(1,1), (2,2), (3,3), (3,4), (4,3), (4,4), (5,5)}. (c) ©{{2,3,4,5}, {1}} ´ 2 = ´ 3 = ´ 4 = ´ 5 ={ 2,3,4,5 } , ´ 1 ={ 1 } The ordered pairs in the equivalence relation on A are: {(1,1,), (2,2), (2,3), (2,4), (2,5), (3,2), (3,3), (3,4), (3,5), (4,2), (4,3), (4,4), (4,5), (5,2), (5,3), (5,4), (5,5)}. 4
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Jennifer Klarfeld – Homework 9 – Section 3.3/3.5 – 8 th Edition 10. Complete the proof of Theorem 3.3.2: Suppose that P is a partition on A and suppose that xQy if there exists C P such that x C and y C . Prove that: (a) Q is symmetric. Suppose that xQy . Then there exists some C P such that x C and y C . It follows that there exists some C P such that y C x C . Therefore, yQx . (d)Q is reflexive on A . Let x A . Then x ¿ X P X , by the definition of a partition of A . This means that x C for some C P . Since for some C P , x C and x C , then xQx . 5
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Jennifer Klarfeld – Homework 9 – Section 3.3/3.5 – 8 th Edition Section 3.5 2. Let A={a,b,c}. Give an example of a relation on A that is (a) antisymmetric and symmetric. R={(a,a), (b,b), (c,c)} This relation is symmetric because for every (x,y) in R, xRy implies yRx. This relation is antisymmetric because its symmetry implies that x=y.
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