# Lecture 001

:book: Proposition: statement proven or dis-proven (a declaration sentence in which it is possible to assign a truth value)

• a proposition may not be true: If n is a non-negative integer, then n^2+42 is a prime.

• or a statement we may not know whether it is true: there are infinite many primes of the form 2^n-1.

• non-example: This statement is false is not a proposition because it is problematic.

• non-example: x^2 >= 8 is not a proposition because we don't know the value of x.

• question is not a proposition

## Various Terminology For True Propositions

:book: Theorem(定理): a major result :book: Corollary(推论): a result which is an immediate consequence of theorem (typically no need to do additional work from Theorem). :book: Lemma(引理): small result, intended to be used in the proof of a theorem, help proof of theorem. :book: Proposition(主张): catch-on term, small truth. (small result less important than theorem.) :book: Conjecture(推测): a proposition that have not been proven but we think it is true.

• For all positive integers n, the greatest common divisor of n^{17}+9 and (x+1)^17+9 is 1

• a counter example happens at: ...

• proven by working with rings...

## Proofs

:book: proof: a proof is a logically valid argument demonstrating the truth of the proposition.

• proving a statement saying that it is improvable is still a prove

Criteria of Proofs

• Truth(logic): logic and steps are mathematically valid. (ie. not making false statements.)

• Clarity(precision): why derive from previous step, identify outside knowledge, sentences of clarification

• Know-your-audience: average student in this course

recitation for tomorrow worksheet for recitation

:question: first quiz?

• write this set, write this symbolically, what does it says. after first: profs and fundamental concepts

:question: difference between unknown / non-proposition?

• see if it take truth value, if so, then proposition

:question: What about a proof that demonstrates a particular statement is unprovable? Would that be classified as a “meta proof”?

• it fall under proposition-meta proof

:question: Quiz will look like?

• first homework will be short answer, other will be proofs

TODO: look at worksheet and be prepared

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