1. A function + : S × S → S for a set S is said...
1. A function + : S × S → S for a set S is said to provide an
associative binary operation on S if r + (s + t) = (r + s) +t for
all r, s, t ∈ S. Show that any associative binary operation + on a
set S can have at most one “unit” element, i.e. an element u ∈ S
such that (*) s + u = s = u + s for all...
Let E = {x + iy : x = 0, or x > 0, y =...
Let E = {x + iy : x = 0, or x > 0, y = sin(1/x)}. Prove that
the set E is connected, but that is not path-connected or connected
by trajectories
and consider B = {z ∈ C : |z| = 1}. prove ( is easy to see it
but how to prove it ) the following questions ¿is B open?¿is B
closed?¿connected?¿compact?
We say that a set A is dense in a metric space (M,d) provided A
bar...
We say that a set A is dense in a metric space (M,d) provided A
bar = M. Show that if two continuous functions f, g :(M,d) tends to
R on a dense set A, then they must agree on M
Let g from R to R is a
differentiable function, g(0)=1, g’(x)>=g(x) for all x>0 and...
Let g from R to R is a
differentiable function, g(0)=1, g’(x)>=g(x) for all x>0 and
g’(x)=<g(x) for all x<0. Proof that g(x)>=exp(x) for all x
belong to R.
Consider the function ?:[0,1] → ℝ defined by ?(?) = 0 if ? ∈
[0,1] ∖...
Consider the function ?:[0,1] → ℝ defined by ?(?) = 0 if ? ∈
[0,1] ∖ ℚ and ?(?) = 1/? if ? = ?/? in lowest terms
1. Prove that ? is discontinuous at every ? ∈ ℚ ∩ [0,1].
2. Prove that ? is continuous at every ? ∈ [0,1] ∖ ℚ
Let S = {A, B, C, D, E, F, G, H, I, J} be the set...
Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...