Finite Simple Group of Order Two

這首歌可以拿來出考卷 xD

※ 引述《cielocot (小島)》之銘言：
: http://www.youtube.com/watch?fmt=18&gl=TW&hl=zh-TW&v=UTby_e4-Rhg
: The path of love is never smooth
: But mine's continuous for you

1. 解釋以下名詞: [8%]
  (1) path
  (2) smooth path
  (3) continuous path
  能否舉出一個反例, 使一個 path is not smooth nor continuous?

: You're the upper bound in the chains of my heart
: You're my Axiom of Choice, you know it's true

2. 填空 [6%]
  (1) A chain is a set paired with a ______ order.
  (2) Upper bound is an element of ______ ordered set s.t. ________
  解釋以下名詞:[9%]
  (1) Axiom of Choice
  (2) Zorn's Lemma
  (3) Well-ordering Theorem

: But lately our relation's not so well-defined
: And I just can't function without you

3. 解釋以下名詞: [4%]
  (1) (binary) relation
  (2) function

: I'll prove my proposition and I'm sure you'll find
: We're a finite simple group of order two
: I'm losing my identity

4. (1) [4%]
   Finite simple group of order two 是個冗句
   請問有幾種刪去單字的方法, 使得刪掉之後得到的句子和原句等價?
   (2) [3%]
   以下那些結構沒有 identity?
   (a) groupoid
   (b) semigroup
   (c) monoid
   (d) quasigroup

: I'm getting tensor every day
: And without loss of generality
: I will assume that you feel the same way
: Since every time I see you, you just quotient out
: The faithful image that I map into

5. 若一個 map f is faithful 則 f is [2%]
   (a) a group homomorphism
   (b) a group representation
   (c) a group presentation
   (d) a group hectormorphism
   (e) a group epimorphism
   (f) a group monomorphism
   (g) a group automorphism

: But when we're one-to-one you'll see what I'm about
: 'Cause we're a finite simple group of order two
: Our equivalence was stable
: A principal love bundle sitting deep inside
: But then you drove a wedge between our two-forms

6. 解釋以下名詞: [16%]
   (1) stable equivalence (of semialgebraic sets)
   (2) principal bundle
   (3) wedge
   (2) 2-form

: Now everything is so complexified
: When we first met, we simply connected
: My heart was open but too dense

7. 解釋以下名詞: [8%]
   (1) complexification (of a real vector space)
   (2) simply connected
   (3) open (in a topological space (X,T) )
   (4) dense

: Our system was already directed
: To have a finite limit, in some sense

8. 解釋以下名詞: [12%]
   (1) Directed set
   (2) Direct System (over a direct set)
   (3) limit (in a category)

: I'm living in the kernel of a rank-one map
: From my domain, its image looks so blue,
: 'Cause all I see are zeroes, it's a cruel trap

9. 解釋以下名詞 [4%]
  (1) kernel (of a linear map)
  (2) rank (of a linear map)

: But we're a finite simple group of order two
: I'm not the smoothest operator in my class,
: But we're a mirror pair, me and you,

10. 填空 [6%]
  (1) An operator is a mapping from one _____ or _____ to another.
  (2) A smooth operator is _______  differentiable.

: So let's apply forgetful functors to the past

11. 給出三個 forgetful functors 的例子 [9%]

: And be a finite simple group, be a finite simple group,
: Let's be a finite simple group of order two
: (Oughter: "Why not three?")

12. 分類 groups of order 3 [4%]

: I've proved my proposition now, as you can see,
: So let's both be associative and free

13. 定義 free group [2%]

: And by corollary, this shows you and I to be
: Purely inseparable. Q. E. D.

14. 給出一個既 purely inseparable 又 separable 的例子 [3%]

: ──A Klein Four original by M. Salomone