Question

Are any of the following implications always true? Prove or give a counter-example.

a) f(n) = Θ(g(n)) -> f(n) = cg(n) + o(g(n)), for some real constant c > 0. *(little o in here)

b) f(n) = Θ(g(n)) -> f(n) = cg(n) + O(g(n)), for some real constant c > 0. *(big O in here)

Answer #1

(n) is nothing but the tighter bound within the two functions.SO lets get started with two points.

**a) f(n) =** (g(n)) -> f(n)
= cg(n) + o(g(n)) for some real constant c > 0.,

Now, considering the above equation f(n) = ((g(n)) is nothing but f(n) = O(g(n)) and f(n) =(g(n)) , so not exactly but in a way we can say f(n) = g(n)

Now, o(g(n)) is nothing but f(n)<g(n) ( here "Little o" means Less Than ) so, by login when we apply to equation

(g(n)) -> f(n) = cg(n) + o(g(n))

f(n) >cf(n) + f(n) (where f(n) = g(n) since **f(n)
=** (g(n)))

So, the statement is false

b)

This question is same as first one but the only difference is big O notation.

Now quickly learn this thing that small o notation is tightly lower bounded and Big O is Lower bound so roughly,

**O(n) means less than or equal to**

**o(n) means less than**

so using your equation,

f(n) = Θ(g(n)) -> f(n) = cg(n) + O(g(n)),

f(n)<= cf(n) + f(n)

Therefore it might or might not be true(Depend on the
cases)

**So rigidly it is False also**

Prove or give a counter example: If f is continuous on R and
differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f
is differentiable on R .

Let f,g be positive real-valued functions. Use the definition
of big-O to prove: If f(n) is O(g(n)), then f2(n)+f4(n) is
O(g2(n)+g4(n)).

Assume that f(n) = O(g(n)). Can g(n)
= O(f(n))? Are there cases where g(n) is
not O(f(n))? Prove your answers (give examples
for the possible cases as part of your proofs, and argue the result
is true for your example).

Prove the statement true or use a counter-example to explain why
it is false.
Let a, b, and c be natural numbers. If (a*c) does not divide
(b*c), then a does not divide b.

NOTE- If it is true, you need to prove it and If it is
false, give a counterexample
f : [a, b] → R is continuous and in the open interval (a,b)
differentiable.
a) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0.
(TRUE or FALSE?)
b) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)
c) If f is reversable, f has no critical point. (TRUE or
FALSE?)
d) If a is a “minimizer”...

either prove that it’s true by explicitly using limit laws, or
give examples of functions that contradict the statement
a) If limx→0 [f(x)g(x)] exists as a real number, then both
limx→0 f(x) and limx→0 g(x) must exist as real numbers
b) If both limx→0 [f(x) − g(x)] and limx→0 f(x) exist as real
numbers, then limx→0 g(x) must exist as a real number

Prove mathematically that if a Turing Machine runs in time
O(g(n)), then it runs in time O(h(g(n))+c), for any constant c
>= 0 and any functions g(n) and h(n) where h(n) >= n.

Prove or give a counter-example:
(a) if R ⊂ S and T ⊂ U then T\ S ⊂ U \R.
(b) if R∪S⊂T∪U, R∩S= Ø and T⊂ R, then S ⊂ U.
(c) if R ∩ S⊂T ∩ S then R⊂T.
(d) R\ (S\T)=(R\S) \ T

True or False, explain. If false, give counter example.
a) if events A and B disjoint then A and B independent.
b) if events A and B independent then A and B disjoint.
c) It is impossible for events A and B to be both mutually
exclusive and independent.

Prove the following statements:
a) If A and B are two positive semideﬁnite matrices in IR ^ n ×
n , then trace (AB) ≥ 0. If, in addition, trace (AB) = 0, then AB =
BA =0
b) Let A and B be two (diﬀerent) n × n real matrices such that
R(A) = R(B), where R(·) denotes the range of a matrix.
(1) Show that R(A + B) is a subspace of R(A).
(2) Is it always true...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 7 minutes ago

asked 10 minutes ago

asked 25 minutes ago

asked 41 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago