### 分类 题解 下的文章

## BZOJ 2612 数学 构造 同余 最短路

## Prob.

给一个集合 $ S, |S| < 5000, s_i < 10^9 $。

$ q < 5000 $ 组询问，问 $ x $ 是否可以表示为 $ \sum_i a_is_i, a_i \ge 0 $.

## LOJ 525

## Prob.

求 $ f(x) = \sum_{i=0}^n a_ix^i, a_i \in [0,k-1] \cap \mathbb{N}, s.t. \forall x \in \mathbb{Z}, f(x) \equiv 0 \pmod k $.

## Answers to Problems from Logic: A Very Short Introduction

看完的第一本 VSI ~就当形式逻辑总算入了门（自称）。

翻译有够差的还是英文好。

没有人在被 treat well 的情况下反思现状，肯定是因为现实的痛苦。

VSI 这套好书迟早被这些译者毁了。

读书感想可能直接写在这里了。有空写写形式逻辑笔记。

答案有空补。会形式逻辑的可以在评论区回复哦~

## Chapter 1

Is the following inference deductively valid, inductively valid, or neither? Why? *Jose is Spanish; most Spanish people are Catholics; so Jose is Catholic.*

## Chapter 2

Symbolize the following inference, and evaluate its validity. *Either Jones is a knave or he is a fool; but he is certainly a knave; so he is not a fool.*

## Chapter 3

Symbolize the following inference, and evaluate its validity. *Someone either saw the shooting or heard it; so either someone saw the shooting it someone heard it.*

## Chapter 4

Symbolize the following inference, and evaluate its validity. *Everyone wanted to win the prize; so the person who won the race wanted to win the prize.*

## Chapter 5

Symbolize the following inference, and evaluate its validity. *You made an omelette, and you don't make an omelette and not break an egg; so you broke an egg.*

## Chapter 6

Symbolize the following inference, and evaluate its validity. *It's impossible for pigs to fly, and it's impossible for pigs to breathe under water; so it must be the case that pigs neither fly nor breathe under water.*

## Chapter 7

Symbolize the following inference, and evaluate its validity. *If you believe in God, then you go to church; but you go to church; so you believe in God.*

## Chapter 8

Symbolize the following inference, and evaluate its validity. *It has always rained, and it always will rain; so it's raining now.*

## Chapter 9

Symbolize the following inference, and evaluate its validity. *Pat is a woman, and the person who cleaned the windows is not a woman; so Pat is not the person who cleaned the windows.

## Chapter 10

Symbolize the following inference, and evaluate its validity, where the level of acceptability is 0.5. *Jenny is clever; and either Jenny is not clever or she is beautiful; so Jenny is beautiful.*

## Chapter 11

The following set of statistics was collected from ten people (called 1-10).

$$ \begin{matrix} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\ \hline \text{Tall} & v & & v & & v & & & & v & \\\\ \text{Wealthy} & v & & v & & v & & v & v & & \\\\ \text{happy} & v & v & & v & v & & & v & v & \\\\ \end{matrix} $$

If *r* is a randomly chosen person in this collection, assess the inductive validity of the following inference. *r is tall and rich; so r is happy.*

## Chapter 12

Suppose there are two illnesses, *A* and *B*, that have exactly the same observable symptoms. 90% of those who present with the symptoms have illness *A*; the other 10% have illness *B*. Suppose, also, that there is a pathology test t distinguish between *A* and *B*. The test gives the correct answer 9 times out of 10.

- What is the probability that the test, when applied to a randomly chosen person with the symptoms, will say that they have illness
*B*? - What is the probability that someone with the symptoms has illness
*B*, given that the test says that they do?

## Chapter 13

You hire a car. If you do not take out insurance, and you have an accident, it will cost you $1,500.

If you take out insurance, and have an accident, it will cost you $300.

The insurance costs $90, and you estimate that the probability of an accident is 0.05. Assuming that the only considerations are financial ones, should you take out the insurance?