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Discrete Structures - Spring 1995 - Exam 2
Symbols: N denotes the Natural Numbers, Z denotes the Integers,
Q denotes the Rational Numbers, and R denotes the Real Numbers.
1. Circle T if the statement is true or F if the statement is false.
T F If A is a set, then AxA is a function.
T F If A is a set, then AxA is an equivalence relation.
T F If the sets A, B, C, and D partition a set X, then the relation
R = (AxA) U (BxB) U (CxC) U (DxD) is an equivalence relation on X.
TFIf f is a 1-1 and ONTO function from A to B, then A = B.
TFIf f is a 1-1 function from A to B, then |A| = |f(A)|.
TFIf f:A → B and g:B → C are functions, then (g ° f ) −1 = (g −1 ° f −1).
T F If n is a positive integer, 1 + 10 + 100 + ... + 10(n−1) = (10n − 1)/9.
T F Let DIV denote the INTEGER DIVIDE operation. If a, b, and p are positive integers
and a ≡ b mop p, then [a − p(a DIV p)] = [b − p(b DIV p)].
TFIf 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.
T F There cannot exist a 1-1 correspondence between N and Q.
2. Rewrite: into summation notation.
3. Let R = {(a,b) | a,b ∈ {1,2,3,4} and a/b < 1}. 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 = 5 and s1 = 15, then si is divisible by 5 for all i > 1.
5. List out the ordered pairs that make up the equivalence relation is induced by the partition
{2}, {1,3}, {0,4,5} of the set {0,1,2,3,4,5}?
6. Find the Domain and Image of the function f = {(a,3),(w,12),(e,2),(s,5),(t,3),(g,2),(v,1)}.
7. Let d be the density function on Σn, the set of all n-long strings and define the relation R on Σn
R = {(s,t) | s,t ∈ Σn and d(s) = d(t)}.
(a) Show R is an equivalence relation on Σn. (b) Describe the partition of Σn induced by R.
8. Prove the function f:R → R given by f(x) = 5x + 3 is a bijection (1-1 and onto).
1
12⋅
---------- 2
23⋅
---------- 4
34⋅
---------- 8
45⋅
---------- 16
56⋅
---------- …512
10 11⋅
----------------++++++
12⋅23⋅34⋅…nn 1+()++++ nn 1+()n2+()
3
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