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Knots of Nots

Some equivalents using negations:

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

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 $\displaystyle{\lim_{x\to a}f(x)=L}$ 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 $\displaystyle{\lim_{x\to a}f(x)=L}$'' 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.

Larry Stout 2001-08-17