Cheap Digital Piano Singapore
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Is your child beginning in trying out learning piano? Buying an actual upright piano would cost a few thousand dollars, and may not be a good idea, as many children may stop learning after a few months. Practicing piano is an excellent way to improve musical skills and even Math skills, as music has been found to be linked to math. Unfortunately practicing piano requires a lot of patience and perseverance (e.g. 1 hour practice every day), in order to reach a decent level like Grade 8 Piano. Hence, many children understandably cannot sustain the necessary practice required, and stop learning, and the money buying the piano (and space in the house) would be wasted.
Research has found that music helps children learn maths! Listening to music in maths lessons can dramatically improve children’s ability in the subject and help them score up to 40 per cent higher in examinations, a new study has found. Source: http://www.telegraph.co.uk/education/9159802/Music-helps-children-learn-maths.html
Buying a keyboard (non-weighted) is also not a good idea, as classical piano requires dynamics (e.g. forte and piano) which means loud and soft contrasts, which is not achievable using a non-weighted keyboard.
The perfect solution is to purchase a digital piano first, and then upgrade to a upright piano if the child maintains interest after a few years. A very good brand for starters would be Yamaha, a Japanese music brand. An added advantage of digital pianos is the ability to use earphones, which would enable one to practice late at night without disturbing the neighbor.
If the child is interested to pursue MEP (Music Elective Programme), this digital piano would come in handy for musical composition in MIDI, and it would have orchestral sounds like Strings, Brass, Drums, etc. Definitely a very useful instrument to have for serious music students as well as the amateur.
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, there is a unique map ![p:[a,b]\to [c.d]](https://s0.wp.com/latex.php?latex=p%3A%5Ba%2Cb%5D%5Cto+%5Bc.d%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
of the form p(x)=mx+k that maps a to c and b to d. This is called the positive linear map of [a,b] to [c,d] because its graph is a straight line with positive slope.
in X as follows: On [0,a] it equals the positive linear map of [0,a] to I=[0,1] followed by f; on [a,b] it equals the positive linear map of [a,b] to I followed by g; on [b,1] it equals the positive linear map of [b,1] to I followed by h. This path 
is path homotopic to 
be the map whose graph is pictured in Figure 51.9 (taken from Munkre’s Book)
. Let’s see why this is so.
is the positive linear map of [0,a] to [0,c], followed by the positive linear map of [0,c] to I, followed by f. This equals the positive linear map of [0,a] to I, followed by f, which is precisely 
is a path in I from 0 to 1, and so is the identity map 
. Since I is convex, there is a path homotopy P in I between p and i. Then, 
is a path homotopy in X between 
.
is exactly the triple product 
and 
.![(g*h)(s)=\begin{cases} g(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ h(2s-1)\ &\text{for }s\in [\frac{1}{2},1] \end{cases}](https://s0.wp.com/latex.php?latex=%28g%2Ah%29%28s%29%3D%5Cbegin%7Bcases%7D++++g%282s%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C++++h%282s-1%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C1%5D++++%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![f*(g*h)(s)=\begin{cases} f(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ (g*h)(2s-1)\ &\text{for }s\in [\frac{1}{2},1] \end{cases} =\begin{cases} f(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ g(4s-2)\ &\text{for }s\in [\frac{1}{2},\frac{3}{4}]\\ h(4s-3) &\text{for }s\in [\frac{3}{4},1] \end{cases}](https://s0.wp.com/latex.php?latex=f%2A%28g%2Ah%29%28s%29%3D%5Cbegin%7Bcases%7D++++f%282s%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C++++%28g%2Ah%29%282s-1%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C1%5D++++%5Cend%7Bcases%7D++++%3D%5Cbegin%7Bcases%7D++++f%282s%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C++++g%284s-2%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C%5Cfrac%7B3%7D%7B4%7D%5D%5C%5C++++h%284s-3%29+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B3%7D%7B4%7D%2C1%5D++++%5Cend%7Bcases%7D++++&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
when c=1/4 and d=1/2. Thus, the these two products are path homotopic, and we have finally proven the associativity of *.
, where 
is the threshold of hearing, which is the smallest sound a human can hear.
be a measure defined on a 
-algebra X.
) is an increasing sequence in X, then
) is a decreasing sequence in X and if 
, then
. A decreasing sequence means the opposite, i.e. 
.
for some n, then both sides of the equation are 
, and inequality holds. Henceforth, we can just consider the case 
for all n.
and 
for n>1. Then 
is a disjoint sequence of sets in X such that
, 
(since (
) is a disjoint sequence of sets)
, we have that 
for n>1, so the finite series on the right side telescopes to become
, so that 
is an increasing sequence of sets in X.![\begin{aligned} \mu (\bigcup_{n=1}^\infty E_n) &=\lim \mu (E_n)\\ &=\lim [\mu (F_1)-\mu (F_n)]\\ &=\mu (F_1) -\lim \mu (F_n) \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++++%5Cmu+%28%5Cbigcup_%7Bn%3D1%7D%5E%5Cinfty+E_n%29+%26%3D%5Clim+%5Cmu+%28E_n%29%5C%5C++++%26%3D%5Clim+%5B%5Cmu+%28F_1%29-%5Cmu+%28F_n%29%5D%5C%5C++++%26%3D%5Cmu+%28F_1%29+-%5Clim+%5Cmu+%28F_n%29++++%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, it follows that
.
, let 
denote the constant path 
mpping all of I to the point x. If f is a path in X from 
to 
, then ![[f]*[e_{x_1}]=[f]](https://s0.wp.com/latex.php?latex=%5Bf%5D%2A%5Be_%7Bx_1%7D%5D%3D%5Bf%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![[e_{x_0}]*[f]=[f]](https://s0.wp.com/latex.php?latex=%5Be_%7Bx_0%7D%5D%2A%5Bf%5D%3D%5Bf%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
be the path defined by 
. ![[f]*[\bar{f}]=[e_{x_0}]](https://s0.wp.com/latex.php?latex=%5Bf%5D%2A%5B%5Cbar%7Bf%7D%5D%3D%5Be_%7Bx_0%7D%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![[\bar{f}]*[f]=[e_{x_1}]](https://s0.wp.com/latex.php?latex=%5B%5Cbar%7Bf%7D%5D%2A%5Bf%5D%3D%5Be_%7Bx_1%7D%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
is a continuous map, and if F is a path homotopy in X between the paths f and f’, then 
is a path homotopy in Y between the paths 
and 
.
is a continuous map and if f and g are paths in X with f(1)=g(0), then
, where h=f*g as defined previously.
.
denote the constant path in I at 0, and we let 
is also a path in I from 0 to 1.
(Straight-line homotopy) Then 
is a path homotopy in X between the paths 
and 
(Lemma 1). Furthermore by Lemma 2, 
which is equivalent to 
.
denotes the constant path at 1, then 
is path homotopic in I to the path i, shows that 
. Then 
is a path in I beginning and ending at 0. The constant path 
is also beginning and ending at 0. Again, because I is convex, there is a path homotopy H in I between 
is path homotopy between 
and 
. Very similarly, we can use the fact that 
is path homotopic in I to 
. It covers the areas of Algebra, Geometry, Counting and Probability, and Number Sense, over 500 examples and problems with fully explained solutions.
and measures should be additive over disjoint sets in X.
for all 
) of sets in X, then
.
, we say it is finite. More generally, if there exists a sequence 
and such that 
for all n, then we say that 
be definied on X by 
, for all 
. We can see that 
be defined by 
, 
if 
. 
defined on B which coincides with length on open intervals. I.e. if E is the nonempty interval (a,b), then 
. This measure is usually called Lebesgue measure (or sometimes Borel measure). It is not a finite measure since 
. But it is 
, we define the product f*g of f and g to be he path h given by the equations![h(s)=\begin{cases}f(2s) &\text{for }s\in [0,\frac{1}{2}], \\ g(2s-1)& \text{for }s\in[\frac{1}{2}, 1]\end{cases}](https://s0.wp.com/latex.php?latex=h%28s%29%3D%5Cbegin%7Bcases%7Df%282s%29+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C+%5C%5C+g%282s-1%29%26+%5Ctext%7Bfor+%7Ds%5Cin%5B%5Cfrac%7B1%7D%7B2%7D%2C+1%5D%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![[f]*[g]=[f*g]](https://s0.wp.com/latex.php?latex=%5Bf%5D%2A%5Bg%5D%3D%5Bf%2Ag%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![H(s,t)=\begin{cases}F(2s,t) &\text{for }s\in[0,\frac{1}{2}],\\ G(2s-1,t)&\text{for }s\in[\frac{1}{2},1]\end{cases}](https://s0.wp.com/latex.php?latex=H%28s%2Ct%29%3D%5Cbegin%7Bcases%7DF%282s%2Ct%29+%26%5Ctext%7Bfor+%7Ds%5Cin%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C%5C%5C+G%282s-1%2Ct%29%26%5Ctext%7Bfor+%7Ds%5Cin%5B%5Cfrac%7B1%7D%7B2%7D%2C1%5D%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
belongs to X for every set E belonging to Y.
the set 
belongs to X.
for every Borel set E.
.
if f and f’ are path homotopic.
and 
are equivalence relations.
. The map F(x,t) is the required homotopy.
. Let F be a homotopy between f and f’. We can then verify that G(x,t) = F(x, 1-t) is a homotopy between f’ and f.
, we show that 
. Let F be a homotopy between f and f’, and let F’ be a homotopy between f’ and f”. This time, we need to define a slightly more complicated homotopy G: X x I -> Y by the equation![G(x,t) = \begin{cases} F(x,2t) &\text{for }t\in [0,\frac{1}{2}],\\ F'(x, 2t-1) &\text{for } t\in [\frac{1}{2}, 1].\end{cases}](https://s0.wp.com/latex.php?latex=G%28x%2Ct%29+%3D+%5Cbegin%7Bcases%7D+F%28x%2C2t%29+%26%5Ctext%7Bfor+%7Dt%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C%5C%5C+F%27%28x%2C+2t-1%29+%26%5Ctext%7Bfor+%7D+t%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C+1%5D.%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
) in M(X,X) such that:
for 
.
for each 
, let 
be the set
.
, let 
.
are disjoint.
on 
is true.
, i.e. 
, thus 
for each 
are also in M(X,X).
, let 
be the “truncation of f” defined by 
be defined similarly. We will work out the proof that 
to 
is measurable.
, and using an earlier corollary that says that if a sequence 
is in M(X,X) converges to f on X, then f is also in M(X,X), we have that 
belongs to M(X,X).
, and hence fg also belongs to M(X,X).
implies 
.
consists of a half-line which is either of the form 
or the form 
. (We will show later that both cases can occur.) Thus, the set will belong to the Borel algebra B which is the 
, then when 
, the set will be 
.
, 
belong to X and the real-valued function 
defined by 
is measurable.
and 
, then we have that 
which is in X since it is the complement of the union of A and 
.
, then 
, which is a union of two sets in X and hence also in X.
and 
when 
, and 
when 
, due to a similar reason as above. Therefore f is measurable!
belongs to X.
belongs to X,
belongs to X,
belongs to X,
belongs to X,
and 
are complements of each other, hence statement (a) is equivalent to statement (b). This is due to one of the properties of 
belongs to X for each n.
, it follows that 
. Thus, (a) implies (c).
, and hence if (c) is true, each 
is in X, and the union of them is also in X (definition for 
, or even “<” and nothing would change!
by Stephen Covey (Highly recommended to read), there are four types of activities:
Mathtuition88 