高中數學 (1)

何謂邏輯
What is logic ? 一種研究思考判斷及推理的科學
Statement (敘述) : In logic a statement is either (a) a meaningful declarative sentence that is either true orfalse, or (b) that which a true or false declarative sentence asserts . Statement_(logic) ; 白話點的說法,即是可以判斷真假的語句. 
What is proposition ? proposition (命題) 即可以判斷語句的 mathematics statement. 
For example , 2 is a prime number . 
Assume above is a proposition P , P :  2 is a prime number . It is a true proposition . 
Another example , if x ∈ R , x ²  + 2x - 1 = 0 , then has no real number solutions.
Basically , a statement must be justified true or false , but it doesn't allow ambiguous (語意模糊) semantic ; otherwise , it is not a statement and justify a statement's true or false is a proposition .         
what is 詭論 ? 

Proposition Calculus : P ⇒Q ; P is predicate (謂詞) 而 Q is qualifier(量詞) . 
Predicate & Qualifier , There is a detail introduction. From the viewpoint of set , P ⊂ Q . 

P, Q 's truth table , just only one is false ; that is , P is true and Q is false , then  P ⇒Q is false ; otherwise others are true . 
反過來說 , 若 P 為真(true) , 而 Q為假(false) , 則  P ⇒Q 為假. 

若充分命題成立,則必要命題必須成立;否則,整個命題為假. 這種命題推論在充分條件成立的前提之下, 則必要條件必然存在; otherwise , 命題是否定.   

For example , 0.3333... 是無理數 . 

Let x = 0.3333... , 10 x = 3.333 ... = 3 + x 則 99x = 3 , x = 3/99  = 1/33 . We revisit the definition of rational : Q = { m/n | m , n ∈ Z , n ≠ 0 } ∴ it is false proposition. 
For example , 
(x-1) ²  + (x-2) ²  = 0  

∵ (x-1) ² ≥ 0 , (x-2) ²  ≥ 0 ∴ (x-1) ²   = 0 and (x-2) ²  = 0 ⇒x = 1 and x =2 ,  So, it is contradictory. On other hands , we may expand above expression as below : 
x ²  - 2x + 1 + x ²  - 4x + 4 = 2x ²  - 6x + 5 , x = 6 ± √36 - 40 / 4 ∉ Q   

^{推論 P, Q 命題 , 即 if 與 only if 的驗證方法是容易的, 假設有兩命題為 P , Q  ; 則若}
P ⇒Q is true and  Q ⇒P is false , then P is sufficient condition (充分條件) , Q is necessary condition (必要條件) ; 反之亦可證. 
但如果 P ⇒Q is true and  Q ⇒P is true , then P, Q 為 充要條件 ( if and only if )   

Wiki Logic in Discrete Mathematics , Discrete Mathematics - Logic