Consider the relation R defined on the set Z as follows: ∀m, n ∈
Z, (m, n) ∈ R if and only if m + n = 2k for some integer k. For
example, (3, 11) is in R because 3 + 11 = 14 = 2(7).
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.
Consider the following recurrence relation defined only for n =
2^k for integers k such that k ≥ 1: T(2) = 7, and for n ≥ 4, T(n) =
n + T(n / 2). Three students were working together in a study group
and came up with this answer for this recurrence: T(n) = n * log2
(n) − n − log2 (n) + 8. Determine if this solution is correct by
trying to prove it is correct by induction.
Given the relation R = {(n, m) | n, m ∈ ℤ, n ≥ m}. Which of the
following statements about R is correct?
R is not a partial order because it is not antisymmetric
R is not a partial order because it is not reflexive
R is a partial order
R is not a partial order because it is not transitive
Let Σ be a finite alphabet with n letters and let R be the
relation on Σ* defined as follows: R = {(u, v): every letter in u
occurs somewhere in v, and every letter in v occurs somewhere in u}
Then R is an equivalence relation with exactly 2n equivalence
classes.
T or F?
Consider the following recursive algorithm for computing the sum
of the first n squares: S(n) = 12 +22 +32 +...+n2 . Algorithm S(n)
//Input: A positive integer n //Output: The sum of the first n
squares if n = 1 return 1 else return S(n − 1) + n* n a. Set up and
solve a recurrence relation for the number of times the algorithm’s
basic operation is executed. b. How does this algorithm compare
with the straightforward non-recursive algorithm...
Assume A = R and the relation R ⊆ A × A such that for x, y, ∈ R,
xRy if and only if sin2 x + cos2 y = 1. Prove that R is an
equivalence relation and for any fixed x ∈ R, find the equivalence
class x
Question: Consider the relation R on A defined by aRb iff 1mod4
= bmod4
a)Construct the diagraph for this relation
b)show that R is an equivalence relation
Part B: Now consider the relation R on A defined by aRb iff a
divides b (Divides relation)
c) Show that R is partial ordering
d) Contruct the hasse diagram for this relation
Consider a relation R (ABCDEFGH) with the following functional
dependencies:
ACD --> EF
AG --> A
B --> CFH
D --> C
DF --> G
F --> C
F --> D
Find minimal cover and identify all possible candidate keys. In
order to receive full credit, please list each step taken and the
rules that you applied.