24/11/10 09:27:29.32 zvgSRz4H.net
>>11 補足
(引用開始)
・では、背理法は？ (qの否定(￢q)) ・ p ⇒ 矛盾 （空集合Φ、 ”・”は積です）
つまり ベン図で P∩Q^- ＝Φ（空集合）
です
・背理法の利点は、証明に使える条件が増えていること
つまり、p ⇒q の証明は、pのみを使って q を導くのに対して
背理法では、pに加えて qの否定(￢q)も使えて、矛盾 （空集合Φ）を導けば良いってことです。この方が楽な場合があるってこと
(引用終り)
下記2点に詳しい説明がある
(参考)
URLﾘﾝｸ(math.libretexts.org)
LibreTexts
Disjoint Sets
This is an instance where proving the contrapositive or using a proof by contradiction could be reasonable approaches. To illustrate these methods, let us assume the proposition we are trying to prove is of the following form:
If P , then T=∅ .
If we choose to prove the contrapositive or use a proof by contradiction, we will assume that T≠∅
. These methods can be outlined as follows:
The contrapositive of “If P , then T=∅ ” is, “If T≠∅ , then ┐P .”
So in this case, we would assume T≠∅ and try to prove ┐P .
Using a proof by contradiction, we would assume P and assume that T≠∅ .
From these two assumptions, we would attempt to derive a contradiction.
One advantage of these methods is that when we assume that T≠∅ , then we know that there exists an element in the set T .
We can then use that element in the rest of the proof.
We will prove one of th the conditional statements for Proposition 5.14 by proving its contrapositive.
The proof of the other conditional statement associated with Proposition 5.14 is Exercise (10).
Proposition 5.14
Let A and B be subsets of some universal set. Then A⊆B if and only if A∩Bc=∅ .
Proof
略
URLﾘﾝｸ(youtu.be)
Proof by Contradiction (full lecture)
Dr. Valerie Hower
2020/11/08
コメント
@monraet
5 か月前
Thanks Dr. However for all your math videos, they are the best