## Q1
Please determine whether the followings are true or false in the universe of real number.
1. $\forall a \exists x(x^2 =a)$
2. $\forall a \exists a(x+a =0)$
3. $\exists a \forall x(x+a =0)$
4. $\forall x \exists a(ax =0)$
5. $\forall x \exists y(x<y)$
6. $\exists x \forall y(x<y)$
7. $\forall x \forall y(x=y\Rightarrow x^2 = y^2)$
8. $\forall x \forall y(x^2=y^2\Rightarrow x = y)$

## Q2
Which one is tautology?
1. $(P\vee \neg P) \wedge Q$
2. $(P\vee \neg P) \vee Q$

## Q3
Using truth tables, show that

$$ \neg( P\rightarrow Q)\iff(P\vee \neg Q).$$

## Q4
Is $(P\vee Q)$and$(\neg Q\rightarrow P)$ logically equivalent?

## Q5
Please show that $P\rightarrow(Q\rightarrow R) \iff (P\wedge Q)\rightarrow R$.

## Q6
Please show that if $A-B\subseteq C$ then $A-C\subseteq B$.

## Q7
Let $A_n = \left\{1, 2, 3\cdots n\right\}$ for $n\in\mathbb{N}$, please evaluate $\cup_{n\in \mathbb{N}}A_n, \cap_{n\in \mathbb{N}}A_n.$

## Q8
Let $p$ and $q$ be prime numbers. Define $A = \left\{p^i, i\in \mathbb{N}\right\}$ and $B = \left\{q^i, i\in \mathbb{N}\right\}$,
show that if $A\cap B\neq \phi$ then $A=B$.

## Q9
Consider $A_n = \left[\dfrac{1}{n}, 1\right]$ please find $\cup_{n\in \mathbb{N}}A_n$, and explain your reason.

## Q10
Consider $A_n = \left[-1, \dfrac{1}{n}\right)$ please find $\cap_{n\in \mathbb{N}}A_n$, and explain your reason.

You may use the [Archimedean property](https://zh.wikipedia.org/wiki/%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E5%85%AC%E7%90%86)