Question

Prove that the isometries is a bijective functions that is in R^2 to R^2.

Answer #1

2) Characterize the following functions in terms of whether they
are injective, surjective, and/or bijective. If possible, find
their inverses.
a. ?:ℝ+ → ℝ+ with ?(?) = ?3
b. ?:ℤ+ → ℤ+ with ?(?) = ?3
c. ?:? → ? with ? being the set of calendar dates and ?(?) being
the day numbered ? in the next month that has that day. For
instance, if ? is January 22, 1981 and ? is March 31, 1993, ?(?) is
February...

Determine which of the following functions are injective,
surjective, bijective (bijectivejust means both injective and
surjective).
(a)f:Z−→Z, f(n) =n2.
(d)f:R−→R, f(x) = 3x+ 1.
(e)f:Z−→Z, f(x) = 3x+ 1.
(g)f:Z−→Zdefined byf(x) = x^2 if x is even and (x −1)/2 if x is
odd.

Let f: A→B be
bijective. Prove
that for each b
in B, there exists
a unique a in
A such that f(a)
= b.

Define f: R (all positive real numbers) -> R (all positive
real numbers)
by f(x)= sqrt(x^3+2)
prove that f is bijective

Prove that 1 + r + r^2 + … + r n = (1 – r^(n +1) )/(1 – r) for
all n ∈ N, when r ≠ 1

Let G be a region in the complex plane. Prove that the set of
functions that are harmonic in G is a vector
space (over R)

Prove the statement " For all real numbers r, if r is
irrational, then r/2 is irrational ". You may use any method you
wish. Be sure to state what method of proof you are using.

Prove that |R^2| = |R| by showing |(0,1) x (0,1)| = |(0,1)| through
cardinality.

Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤ (a1)2 + (a2)2 +
... + (an)2.
Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).
Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s...

6. Let G = S5 the group of bijective maps on Ω5 = {1,2,3,4,5}.
Let H = {σ ∈ G : σ(5) = 5}.Let K be a subgroup of G. Prove that KH
= G if and only if 5 divides |K|.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 10 minutes ago

asked 19 minutes ago

asked 33 minutes ago

asked 35 minutes ago

asked 38 minutes ago

asked 45 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