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Discrete Structures - Fall 2000 - Exam 2
Symbols: N denotes the Natural Numbers, Z denotes the Integers, Q denotes the Rational
Numbers, and R denotes the Real Numbers.
1. (20 pts.) Circle T if the statement is true or F if the statement is false.
T F If A is a non-empty set, then AรA is the largest Equivalence Relation on A.
TFIf c is a real number, and {sk} and {tk} are sequences of real numbers, then
.
T F The Equivalence Relation induced by the partition {{1,3,5,7},{2,6,8},{4}} of
{1,2,3,4,5,6,7,8} has 8 elements.
T F There cannot exist an ONTO mapping from {1,2,3,4} to {1,2,3}.
TFIf f:A โ B is a function, then |A| = | f (A)|.
TFIf f:A โ B and g:B โ C are functions, then (g ยฐ f ) โ1 = (f โ1 ยฐ g โ1).
TFIf n is a positive integer, 1 + ฯ + ฯ2 + ... + ฯ(nโ1) = (ฯn โ 1).
T F The Weak and Strong Forms of Mathematical Induction are equivalent.
T F If H is the Hamming distance function, d is the density function, and 0 is the
all-zero string, then H(s,0) = d(s) for all binary strings, s.
TF|N| = |Q|.
2. Rewrite: as a summation from 6 to 16.
3. Let R = {(a,b) | a,b โ {1,2,3,4,5} and (a2 + b2) is prime}. Graph R.
4. Do 1 of the following 2 induction proofs:
Prove: .
or
Prove: If si is a recursively generated sequence given by si = si-1 + si-2,
with s0 = โ5 and s1 = 5, then si is divisible by 5 for all i > 1.
5. Show that (g ยฐ f ) โ1 = (f โ1 ยฐ g โ1) for the functions on {1,2,3,4,5} given by:
f = {(1,3),(2,2),(3,5),(4,1),(5,4)} and g = {(1,2),(2,3),(3,4),(4,5),(5,1)}.
6. Find the Domain and Image of the function f = {(d,3),(i,12),(s,2),(c,5),(r,3),(e,2),(t,1)}.
Domain = { } Image = { }
7. Let R be the relation R = {(a,b) | a,b โ Z and a โก b mod 5}.
(a) Show R is an equivalence relation on Z. (b) Describe the partition of Z induced by R.
8. Prove: For all a,b โ R, the function f :R โ R given by f(x) = 15x + 13 is a bijection (1-1
and onto).
csktk
+()
k0=
n
โcs
k
k0=
n
โ
โโ
โโ
โโ
โโ
tk
k0=
n
โ
โโ
โโ
โโ
โโ
+=
1
12โ
---------- 2
23โ
---------- 4
34โ
---------- 8
45โ
---------- 16
56โ
---------- โฆ1024
11 12โ
----------------++++++
12โ
23โ
34โ
โฆnn 1+()++++ nn 1+()n2+()
3
--------------------------------------=
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