离散数学Exercises

1.List all subsets of {JA V A, PASCAL, C++}
2.Prove that if A ⊆ B and C ⊆D, then A ∩ C ⊆B ∩ D
3.A, B and C are sets, prove or disprove the following statements.
(1) If A – B = A – C, then B = C
(2) If A⨯ B = A⨯ C, A ≠∅, then B = C
4.Write each of the following statements in terms of propositional variables, predicates,
quantifiers and logical connectives. You can choose any propositional variables and predicates freely.
(1) If I like the course or the teacher, I will attend the class. (Statement and its negation)
(2) For all students of our school, someone studies hard and has good score; someone
studies hard and has not good score.
5.Prove that the following is a tautology using truth table
(1) ~()
p p q
⇒⇒(2) (()())()
p q q r p r
⇒∧⇒⇒⇒
6.Let A and B are square matrices. Show that if AB=BA, then (AB)n=A n B n
1. Let A={a,b,c,d,e}, a relation R on A is {(a,b), (b,b), (b,c), (b,d), (c,d), (d,d), (d,e), (e,d)}.
(1) Give the digraph and matrix of relation R;
(2) Compute R2, reflexive closure r(R) and symmetric closure s(R).
2. Let A= {a1, a2, a3, a4, a5}, and the diagram of a relation R on A is shown in Fig.1.
Fig. 1
(1)Give the in-degree and the out-degree of each vertex
(2)Find R∞ for the relation R using Warshall’s algorithm.
3. Let S Z+ and A=S×S. Define the following relation R on A
(a,b) R (a′,b′) if and only if ab′ = a′b
(1) Show that R is an equivalent relation;
(2) Let S={1,2,3,4,5,6,7,8,9}, compute the equivalent class [(2,4)].
1. Let f(n)= 3n 3-5n 2+8 and g(n)=n 3 be defined for positive integer n. Show that f and g have the same order.
2. Let A={1, 2, 3, 4, 5, 6} , and let
121 2 3 4 5 6 1 2 3 4 5 6,3 4 1 2 6 5 2 3 1 5 4 6p p ⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭
Compute
(1)11
p -
(2) 212()p p p 3. Let function f(x, y)=( x+3y, 2x+y), (x, y) in R×R, prove that f is bijection.
1. Determine the Hasse diagram of the relation on A = {1, 2, 3, 4, 5} whose matrix is shown below.
1 0 1 1 10 1 1 1 10 0 1 1 10 0 0 1 00 0 0 0 1⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭
2. Find two different topological sorting of the poset whose Hasse diagram is shown in Fig. 2.
Fig. 2
3. Let A be a finite nonempty poset with partial order ≤.
(1) Show that A has at least one maximal element;
(2) Is there a greatest element for all possible A? If not, show a counterexample.
4. Let A={2, 4, 5, 6, 8, 10, 12, 20, 120}, R is the relation of divisibility on A.
(1) Draw the Hasse diagram of the poset <A, R>;
(2) Find all the minimal elements, the maximal elements, the least element and the greatest element of the poset <A, R> if they exist;
(3) Let B = {2, 4, 6}, find the upper bound, the lower bound, the least upper bound and the greatest lower bound of B if they exist.
Chapter 7 & 8
1.Construct the tree of the algebraic expression
( 7 + (6 - 2) ) – ( x- (y - 4) )
e Fleury’s algorithm to produce an Euler circuit for the graph in Fig. 3
Fig. 3
e the labeling algorithm (Ford-Fulkerson’s) to find a maximum flow for the following
transport network in Fig. 1. Use of figures is required to show the variety of flows during the procedure.
Fig.1 transport network
e Kruskal’s algorithm to find a minimal spanning tree of graph in Fig. 2. The sequence of
edges-selecting is ordered to be shown up.
C D。