MIT-计算机科学中的数学 Lec1-3

课程链接：https://www.bilibili.com/video/BV1zh41167Uy?from=search&seid=3400639910032477988&spm_id_from=333.337.0.0

Lec-1

Def: A mathematical proof is a verification of a proposition by a chain of logical deductions from a set of axioms

Def: A proposition（命题） is a statement that is either true or false
Ex: 2+3=5

Def: A predicate（谓词） is a proposition whose truth depends on a value of a variable
Ex: the Goldbach’s conjecture(哥德巴赫猜想)
the Riemann hypothesis（黎曼假设）
the Poincare conjecture（庞克类猜想）

Def: implication（隐含）： An implication p→q is true if p is false or q is true（当p为假或q是为真，p→q为真）
if and only if （当且仅当）↔ ：p→q & q→p
Truth Table（真值表）：

p	q	p→q	q→p	p↔q
T	T	T	T	T
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

p if and only q is true when they’re both true or both false

​ Examples of something that’s NOT A PROPOSITION:

• this statement is false
• a question

Def: axiom(公理): An axiom is a proposition that is assumed to be true
Ex: if a=b and b=c, a=c

​ two guiding principles to axioms:

1. consistent(一致)
2. complete(完整)

Def: a set of axioms is consistent if no proposition can be proved to be both true and false
a set of axioms is complete if it can be used to prove every proposition is either true or false

Lec-2

indirect proof(间接证明)：

Proof by Contradiction(反证法):

To proof P is true, we assume P is false (┐P is T),then use that hypothesis to derive a falsehood or contradiction
if ┐P →F is true,P is true.

Ex: Thm: $\sqrt{2}$ is irrational

Pf:(by cont)
Assume for purpose of contradiction that $\sqrt{2}$ is rational
→$\sqrt{2}$ = a/b (a fraction in lowest terms: a and b have no common divisors)
→ 2= $a^2$/$b^2$
→ 2*$b^2$=$a^2$
→ a is even(2|a) a是偶数
→ 4|$a^2$
→ 4|2*$b^2$
→ 2|$b^2$
→ b is even
→a/b is not a fraction in lowest terms(contradiction)
→$\sqrt{2}$ is rational

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