MAT1830 - Discrete Mathematics for Computer Science Assignment #3
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MAT1830 - Discrete Mathematics for Computer Science
Assignment #3
(1) Let E be the binary relation on the set {a,b,c,d,e,f,g,h,i} pictured below. Write down whether E is reflexive, symmetric, antisymmetric, transitive. When E does not have one of these prop- erties give an example of why not.
(2) Let R be a binary relation on Z × Z defined by
(w,x)R(y,z) if and only if w ≤ y and x ≡ z (mod 3).
Is R reflexive? Is R symmetric? Is R antisymmetric? Is R transitive?
(3) Let S be the equivalence relation on {X ∈ P({1, 2, 3, 4, 5}) : |X| = 3} defined by YSZ if and only if the sum of the elements of Y is equal to the sum of the elements of Z . Write down the equivalence classes of S .
(4) A positive integer n is chosen and a relation D is defined on the set {x ∈ N : x divides n} by yDz if and only if y divides z .
(a) Draw the Hasse diagram for D when n = 54.
(b) Suppose we know that n ≥ 7 and that D is not a total order relation. What is the least value that n could have?
(c) Suppose we know that n is even, n ≤ 100 and that D is a total order relation. What is the greatest value that n could have?
(d) What is the least positive integer n for which the diagram below is the Hasse diagram of D (for some choice of a,b,c,d,e,f,g)?
2023-05-30