Knots of Nots

Some equivalents using negations:

1. $${\lnot(p \wedge q) \equiv (\lnot p)\vee \lnot q}$$
2. $${\lnot(p \vee q) \equiv \lnot p\wedge \lnot q}$$
3. $${\lnot(p \rightarrow q) \equiv p\wedge \lnot q}$$
4. $${\lnot \forall_x \phi(x)\equiv \exists_x \lnot\phi(x)}$$
5. $${\lnot \exists_x\phi(x) \equiv \forall_x\lnot \phi(x)}$$

Proposition 7   The equivalences 1-3 above are valid: both sides have exactly the same truth values.(3 Points)

(3 Points )

Problem 8   The definition of $${\lim_{x\to a}f(x)=L}$$ is

For any $\epsilon > 0$ there is a $\delta >0$ such that for any $x$, if $0<\vert x-a\vert<\delta$, then $\vert f(x)-L\vert<\epsilon$.

1. Write this formally using $\phi(x,\delta)$ for $0<\vert x-a\vert<\delta$ and $\psi(x,\epsilon)$ for $\vert f(x)-L\vert<\epsilon$.
2. Use the rules for negation above to write the formal expression for ``It is not the case that $${\lim_{x\to a}f(x)=L}$$'' with the negations as far in as possible.
3. State (in English) what it means to say that a function $f$ does not have a limit at $a$, again moving the negations in as far as possible.

(6 Points )

Total for section: 37.