在中三下华文课本中，有一篇题目为《乌敏岛》的课文。课文介绍了乌敏岛的自然风光和纯朴的人文环境。在文章的最后，作者写道“如果能’偷得浮生半日闲’，何不暂时摆脱现实的束缚，和三五好友到乌敏岛游玩呢？”。
“偷得浮生半日闲”出自于唐代诗人李涉的七言绝句《题鹤林寺僧舍》。全文如下：
终日昏昏醉梦间，忽闻春尽强登山。
因过竹院逢僧话，偷得浮生半日闲。
大意是：作者整日昏昏沉沉处于醉梦之中，消磨人生。忽然有一天才意识到春天就要过去了，于是勉强去爬山。在游览寺院的时候碰到一位高僧，便与其闲聊，难得在这纷纷扰扰的世事中获得片刻的清闲。
这首诗的创作背景是李涉官途不顺，被皇帝贬官后又流放到南方，所以其情绪消极终日昏昏沉沉。而在一次偶然机会，登山之时偶遇高僧，闲聊之中，不料解开了苦闷的心结，化解了世俗的烦扰，使得自己心情得以放松。
再回到课文《乌敏岛》，从课文的开始，作者就强调“踏上乌敏岛，映入眼帘的是一幅和繁忙市区截然不同的景象”。市区的人们熙熙攘攘，为生活而忙于奔走，有很多世事的烦扰。而乌敏岛却是一个别样的世界，这里没有喧嚣，人们的生活简单平静而又质朴。来到这里，看看美不胜收的风景，体验淳朴宁静的生活，相信你也会暂时忘记现实生活的烦恼和忧愁。
“偷得浮生半日闲”，人生漂浮不定，难得半日的清闲。
![\mathbb{Z}[x]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5Bx%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. We can show that it is not a PID and hence also not a Euclidean domain.![<2,x>=\{ 2f(x)+xg(x)\vert f(x), g(x) \in \mathbb{Z} [x]\}](https://s0.wp.com/latex.php?latex=%3C2%2Cx%3E%3D%5C%7B+2f%28x%29%2Bxg%28x%29%5Cvert+f%28x%29%2C+g%28x%29+%5Cin+%5Cmathbb%7BZ%7D+%5Bx%5D%5C%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.![<2,x>=<p(x)>=\{ f(x)p(x)\vert f(x)\in \mathbb{Z}[x]\}](https://s0.wp.com/latex.php?latex=%3C2%2Cx%3E%3D%3Cp%28x%29%3E%3D%5C%7B+f%28x%29p%28x%29%5Cvert+f%28x%29%5Cin+%5Cmathbb%7BZ%7D%5Bx%5D%5C%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
, hence 
.
but 
. (contradiction!)
such that 
, where d is the Euclidean function. By the well-ordering principle, every non-empty set of positive integers contains a least element, hence b exists.
be any element in I. 
such that 
, with either r=0, or d(r)<d(b). (This is the property of Euclidean domain.)
be a subgroup of index 2.
and 
.
, then 
, and 
, hence left coset equals to right coset.
, then 
(set minus), and also 
, thus left coset also equals to right coset.
to prove will be quite tricky and convoluted, as seen 
, also known as 
(easier to type).
by 
.
.
.
.
is trivial.
. Consider 
such that 
, 
, 
, …, 
.
.
. This is because 
. Hence if 
is the order of 
, i.e. 
, then 
, which implies that 
which implies that 
. Since the order of 
.
and thus we may take 
is surjective.
by Professor Bartle
-almost everywhere if there exists a subset 
with 
such that the proposition holds on the complement of N. 
, which is discussed in a 
are said to be equal 
when 
, for some 
with 
, 
, 
, where 
is the threshold of hearing, which is the smallest sound a human can hear.
. 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.
-algebra X of subsets of X such that
for all 
is any disjoint sequence (
) 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 
) such that 
for all n.
.
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)
Mathtuition88 