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Discrete Structures - Exam 2 - Spring 1994
1. Let f:A → A be a function and define the relation R on A to be:
R = {(a,b) | a,b ∈ A and f(a) = f(b)}. Prove that R is an Equivalence Relation.
2. Using Mathematical Induction, prove .
3. Rewrite 13 - 13(25) + 13(252) - 13(253) +...- 13(2521) in summation form.
4. Calculate .
5. Let Σ = {0,1} be an alphabet and consider the functions
H:Σn×Σn→Zn, d:Σn→Zn and L:Σn→Zn defined, for any strings s,t in Σn as:
H(s,t) = the number of positions where s and t disagree
(Hamming Distance Function),
d(s) = the number of 1’s in string s (Density Function),
L(s) = the number of characters in string s (Length Function).
Calculate: (a) H(101100111,111001101) (b) d(101100111) (c) L(θ)
6. Rewrite the following as a summation over j using the change of variables
j = k - 2:
7. Evaluate
8. Consider the sequence: si = si-1 + si-2 + si-3 with s0 = 1, s1 = 3, and s2 = 1. Use the Strong Form
of Mathematical Induction to prove that si is odd for all i ∈ N.
9. Prove that each of the following functions is a one-to-one correspondence or show a
counterexample to why it is not:
(a) F:R → R defined by F(x) = 5x + 1
(b) G:R → R defined by G(x) = 3x2 - 5
10. Let A = {0,1,2,3,4} and R be the relation of congruence modulo 2 on A; that is,
R = {(x,y) | x,y ∈ A and 2|(x - y)}.
(a) Draw the directed graph of R.
(b) Describe the partition induced on A by the equivalence relation R.
7m
m0=
p
∑7p1+ 1–
6
---------------------=
1
k
---1
k1+
------------–
⎝⎠
⎛⎞
k1=
10
∑
nk–2+()
k1=
n2+
∑
k!
kk 1–()!
---------------------
k1=
50
∑
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