MA 225 Test # 1 Spring, 1999 Bishir
 

1.      In this problem, the universe is the set of all lawyers in the world. Let

                          U(x) denote the statement "lawyer x is a U.S. citizen",

       (b)    Not every criminal lawyer is a judge.

       (c)    Every criminal lawyer is not a judge.

       (d)    Some criminal lawyers are neither U.S. citizens nor judges.

       (e)    All judges are U.S. citizens or no judges are U.S. citizens.
 
 

2. Symbolize a denial of ($ x) ( ~P(x) Ú ~Q(x) ) which does not use the

symbol ~. Show your steps clearly.
 
 

3. In this question, the universe is the set of all integers. For each of the

following, indicate whether it is TRUE, or FALSE. Be sure to give reasons

for your answers.

(a) (" x) ( x² > 0 ) .

(b) (" x) ($ ! y) ( x + y = 0 ) .

(c) ($ ! x) (" y) [ (y² > 0) ? (x 1 y) ] .
 
 

Work two of problems 4-6. (If you hand in all three, I will grade

#4 and #5 .)
 
 

4. Write a careful proof of: If m is an odd integer, then m² - 5 is even .
 
 

5. Prove or disprove : If r is a negative rational number, then there

exists a rational number n such that r· n = 3 .
 
 

6. Prove that for any real numbers a and b, |a - b| = |b - a| .
 
 

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MA 225 Test # 2 Spring, 1999 Bishir
 
 

Problem 1 is worth 16 points; Problems 2-4 are 28 points each. To receive full credit, give reasons for your claims, and use proper sentence structure and punctuation.
 
 

1. (a) Give a careful statement of the Principle of Mathematical

Induction (PMI) as it relates to a sequence of statements

P(1), P(2), P(3) ... .

(b) Give an inductive definition for the sequence of numbers

ao = 1, a1 = 2, a2 = 22, a3 = 23, a4 = 24, ... .
 
 

2. Prove that, for all integers x, x2 - x + 3 is odd. (Work from

basic definitions of even and odd; you cannot use any 'known'

theorems.)
 
 

3. Prove either (a) or (b), but not both, using Mathematical

Induction.

(a) For n = 3, 4, 5, ... , 5n > 4n + 3n .

(b) For n = 0, 1, 2, ... , 30 + 31 + 32 + ... + ... + 3n = (3n+1 - 1)/2 .
 
 

4. As in # 3, use induction to prove either (a) or (b), but not both.

(a) For all natural numbers n, 6n -1 is divisible by 5 .

(b) For all natural numbers n,

n + (n+1) + (n+2) + (n+3) + (n+4) is divisible by 5.
 
 

Bonus (10 points)

For the statement you proved in # 4, give another, different

proof that does not use induction.
 
 

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MA 225 Test # 3 Spring, 1999 Bishir
 
 
 
 

Respective Point Values for Problems 1-4 are

25 (5-20), 30 (15-15), 22, 28 (8-20).
 
 

1. (a) What is the definition of a subset of a set B?

(b) Prove that for any sets A and B, the difference A-B is a subset

of A.
 
 

2. (a) What is the definition of

(i) a relation from set A to set B?

(ii) a function from set A to set B?

(iii) the inverse of a relation Q?

(b) A relation T is symmetric and transitive, and contains the

ordered pair (3,1). Prove that T cannot be a function.
 
 

3. Work either part (a) or part (b), but not both.

(a) Prove that a relation Q is symmetric if and only if Q = Q-1.

(b) In this problem, X and Y denote subsets of some universal set U.

Relations A and S are defined on U as follows --

(X,Y) ŒA iff X and Y have at least two elements in common;

(X,Y) ŒS iff X is a subset of Y.

Prove that the compositionS° A is actually the same relation

as A.
 
 

4. For this problem, I is the interval [-1, 0] on the real line, and

the function f : IÆ ¬ is defined by

f = {(x,y) ŒI¥ ¬ : y = x2 } .

(a) What are the domain and range of f-1 ?

(b) Prove that f-1 is a function and write it the form

f-1 = {(x,y) ŒSomething : y = Stuff}

where you fill in the proper expressions for Something

and Stuff).