Midterm Examination for Discrete Mathematics
姓名:_______________
學號:________________
2009/12/01
成績:____________
1 (39 pts)
(a) The coefficient of x9 in (2-x)19 is ______________.
(b)  0 ≤ k ≤ 1000 (-1)k3k 21000-k = ________________.
(c) There are _______ non-negative integer solutions to the equations x1 +x2 +
x3 + x4 = 17.
(d) There are _______ integer solutions to the equations x1 +x2 + x3 + x4 ≤ 17,
where x1≥ -4, x2 ≥ 2, x3≥ 7 and x4 ≥ 0.
(e) gcd(500, 284) = _______ = ____x 500 + ____ x 284.
(f) The next 3-combination of {3,5,6} in lexicographical order in the set
{1,2,3,4,5,6} is ___________________.
(g) The next permutation of 634215 is ___________.
(h) Let A = p1  (p2  (p3 (p4 (p5 p6))))) be a propositional formula.
Then there are _______ rows in the truth table for A in which it is true.
(i) There are _____ functions from {1,2,..,10} to {1,…,12}, among which
_________ functions are increasing and _____ are strictly increasing.
2 [10 pts] Which of the following statements are true:
(a). 2x2+ 100x = O(x3).
(b). Every day has 24 hours only if there are two Sundays in a week.
(c).  xy P(x,y)  yx P(x,y) is a valid sentence
(d). If n ≥ m≥ 0, then P(n,m) = C(n,m) x m!.
(e). ~( p /\ q) and (p  ~q) are logically equivalent.
3 [10pts] Translate the following sentences into logical expressions using predicates,
quantifiers and logical connectives.
(a). John likes Mary. [3pts]
(b). Every student in the class owns a car. [3pts]
(c). x is a composite number if it is a product of two other numbers. [4pts]
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4. (6 pts) Identify the error line (or lines) and give your reason in the following
supposed proof of x (P(x)/\Q(x) ) from the premise xP(x) /\ xQ(x).
1.
2.
3.
4.
5.
6.
7.
Ans:
xP(x) /\ xQ(x)
xP(x)
P(c)
xQ(x)
Q(c)
P(c)/\Q(c)
x (P(x)/\Q(x) )
Premise
Simplification from (1).
Existential Instantiation from (2)
Simplification from (1)
Existential Instantiation from (4)
conjunction from (3),(5)
Existential generalization from (6)
Line ___________ is(are) incorrect since
_______________________________________________________________
5. [20pts] Let m1,m2,…,mk ( k > 1) be a sequence of k positive numbers such that
gcd(mi,mj) = 1 for all 1 ≤ i < j ≤ k. Suppose we have found k numbers: x1,x2,…,xk
such that for each 1≤ i ≤ k, xi is a solution of the system of congruence equations:
xi  1 (mod mi) and xi  0 (mod mj) for 1≤ j ≤ k and j i.
(a). Show that for the system of congruences:
x  a1 (mod m1), x  a2 (mod m2),…,x  ak (mod mk),
where a1,…,ak are k integers, x = 1≤ t ≤ k at xt is a solution.
Now let m1,m2,m3 be 11,21 and 31,respectively and a1 a2 and a3 are 5, 7 and 14. You
are now asked to apply the above theorem to solve the system of congruence
equations:
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x  5 (mod 11), x  9 (mod 21) and x  14 (mod 31)
(b). What is the corresponding x1 ,x2 and x3 in the range [0, 11*21*31-1]. [6pts]
(c). Find the unique solution x of the above congruences in the range [0,
11*21*31-1]. [6pts]
6. (25 pts) Let PV={p,q,r,…} be a nonempty set of logical variables and OP = {/\, ~ }
is a set of two logical connectives. The set W of well-formed logical formulas
(abbreviated wff) is a set of strings whose symbols come from the alphabet PV 
OP {(,)}, and is defined inductively as follows:
Basis : If p ∈ PV is a logical variable, then p ∈W is a wff.
Closure: If A and B are two wffs, then so are (A/\B) and ~A.
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(a) Find 3 wffs which are not logical variables and do not appear in this sheet.
[3pts]
(b) Assume p, q and r are logical variables. Explain why (p /\ ~ (q/\ p)) is a wff
while (p /\ q /\ r) is not. [6pts]
(c) Define a recursive function f: Wff  N such that f(A) is the number of
occurrences of logical variables appearing in A. So, for instance, if A is (p /\
~ (q/\p)), then f(A) = 3 since A uses two copies of logical variables p and one
logical variable q. [4pts]
(d) Similarly define the function g : Wff  N recursively such that g(A) is the
number of occurrences of logical connectives used in A. So, for example, if A
is (p /\ ~ (q/\ p)), then g(A) = 3 ( since A uses two ‘/\’ and one ‘~’). [4pts]
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(e)
Prove by structural induction that for all wff A, f(A)  g(A) + 1. [8pts]
7. [10 pts] Show that there exists a onto mapping from N to N2 (=def {(a,b) | a,b ∈
N}). Hence the set N2 is countable.
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