4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 =
?2}. Prove that ? is an...
4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 =
?2}. Prove that ? is an equivalence relation, and
concisely characterize how its equivalence classes are different
from simple real-number equality.
Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) =
sin(?? + ?? + ?2 ).
1. Find a bound for the directional derivative of ? in any
direction, i.e. find a constant ? such that |???(?, ?)|
≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ 2 with
|?| = 1.
Define a function ?∶ ℝ→ℝ by
?(?)={?+1,[?] ?? ??? ?−1,[?]?? ????
where [x] is the integer part function. Is ? injective?
(b) Verify if the following function is
bijective. If it is bijective, determine its inverse.
?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)
Topology question:
Show that a function f : ℝ → ℝ is continuous in the ε − δ
definition of continuity if and only if, for every x ∈ ℝ and every
open set U containing f(x), there exists a neighborhood V of x such
that f(V) ⊂ U.
4.Consider events A and B, neither of which are impossible
events. Prove the following
statements.
A)? ? ? = 1 − ?(?|?)
B)If ? ? ? ? = ? ? ∩ ? , then A and B must be
independent events.
Use induction to prove that 2 + 4 + 6 + ... + 2n = n2 + n for n
≥ 1.
Prove this theorem as it is given, i.e., don’t first simplify it
algebraically to some other formula that you may recognize before
starting the induction proof.
I'd appreciate if you could label the steps you take, Thank
you!
Prove these scenarios by mathematical induction:
(1) Prove n2 < 2n for all integers
n>4
(2) Prove that a finite set with n elements has 2n
subsets
(3) Prove that every amount of postage of 12 cents or more can
be formed using just 4-cent and 5-cent stamps