Sheet7 - McMaster University Department of Computing and...

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Unformatted text preview: McMaster University Department of Computing and Soft- ware Dr. W. Kahl COMP SCI 1FC3 Exercise Sheet 7 COMP SCI 1FC3 Mathematics for Computing 6 February 2009 Exercise 7.1 Partial Orders (a) Show that every totally ordered set is a lattice. (b) Show that every non-empty finite lattice has a least element and a greatest element. (c) Give one example each of an infinite lattice with: (1) neither a least nor a greatest element (2) a least element, but no greatest element (3) a greatest element, but no least element (4) both a least element and a greatest element (d) For each set S among the following sets of rational numbers, prove or disprove that the poset ( S , ) is well-founded: (1) { n + 1 m | n IN m + IN } (2) { n- 1 m | n + IN m + IN } (3) { n- k m | n + IN k IN m + IN k < m } Exercise 7.2 Equivalence Relations (a) If : A A is an equivalence relation and F : A A is total and F holds, then F ; = ....
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Sheet7 - McMaster University Department of Computing and...

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