Discrete Structures - Examination 2 - Spring 2002
1. (20 points) Circle T of the corresponding statement is True and F if it is False:
T F 1 + 5 + 52 + 53 + ... + 5100 = 4(5101 − 1).
T F If A is a non-empty set, then A × A is an equivalence relation on A.
T F If f is a function whose image and range are the same set, then f is ONTO.
T F If f:A → A is a ONE-TO-ONE function then |A| = |f(A)|.
T F If A is a non-empty set, then ∅ is the smallest relation on A.
T F The relation {(1,2),(3,4),(5,6),(7,8)} is a function on the set {1,2,3,4,5,6,7,8}.
T F H(10011001,00000000) = d(10011001).
T F If f:A → B is a function and iA:A → A and iB:B → B are the identity functions on A and B,
respectively, then (iB ° f ) = ( f ° iA) .
T F If a binary string s of length n has d(s) = k, then it has (n − k) zeroes.
T F The Principle of Weak Mathematical Induction is logically equivalent to the Principle of Strong
Mathematical Induction.
2. (10 points) What equivalence relation is induced by the partition { {a},{b},{c},{d},{e} }?
3. (5 points) Write as a summation ranging from i = 5 to 22.
4. (20 points) Let R be the relation on the Integers given by R = {(a,b) | (a − b) = 5k for some integer, k}
a. Show that R is an Equivalence Relation. b. Find [0]?
5. (20 points) Let f:R → R be the function f(x) = 5 − 9x.
a. Show that f is ONE-TO-ONE. b. Show that f is ONTO. c. Find f −1(x).
6. (5 points) Let f = {(0,8),(1,5),(2,6),(3,7),(4,9)} and let g = {(5,1),(6,4),(7,3),(8,2),(9,0)}. Find f−1°g−1.
7. (20 points) Prove 1 of the following 2 statements using the indicated method:
a. Using Strong Induction, show that if n is an integer greater than 1, then n is divisible by a prime number.
b. Using Weak Induction, show that , for all integers n > 1.
a5+()
12
13
--------------------- a6+()
13
24
----------------------a7+()
14
35
--------------------- …a22+()
29
1820
------------------------++++
ii 1+()
i1=
n
∑nn 1+()n2+()
3
--------------------------------------=
Docsity.com