Conjunction and Implication

Theorems
$$ $$ $$ $$

Alternative rendition
They can alternatively be rendered as:

$$ $$ $$ $$

They can be seen to be logically equivalent to the forms above.

The following can also be seen to hold:

$$ $$

which follow immediately from
 * $$\left({p \and \neg q}\right) \iff \left({\neg \left({p \implies q}\right)}\right)$$

and the rule of simplification.

Proof by Natural Deduction
By the tableau method:

Proofs using the LEM
The remaining proofs depend (directly or indirectly) on the Law of the Excluded Middle.

Comment
Note that the Modus Ponendo Tollens:
 * $$\neg \left({p \and q}\right) \dashv \vdash p \implies \neg q$$

can be proved in both directions without resorting to the LEM.

All the others:


 * $$p \and q \vdash \neg \left({p \implies \neg q}\right)$$
 * $$p \implies q \vdash \neg \left({p \and \neg q}\right)$$
 * $$p \and \neg q \vdash \neg \left({p \implies q}\right)$$

are not reversible in intuitionist logic.

Proof by Truth Table
We apply the Method of Truth Tables to the propositions in turn.

As can be seen by inspection, in all cases the truth values under the main connectives match for all models.

$$\begin{array}{|ccc||ccccc|} \hline p & \and & q & \neg & (p & \implies & \neg & q) \\ \hline F & F & F & F & F & T & T & F \\ F & F & T & F & F & T & F & T \\ T & F & F & F & T & T & T & F \\ T & T & T & T & T & F & F & T \\ \hline \end{array}$$

$$\begin{array}{|ccc||ccccc|} \hline p & \implies & q & \neg & (p & \and & \neg & q) \\ \hline F & T & F & T & F & F & T & F \\ F & T & T & T & F & F & F & T \\ T & F & F & F & T & T & T & F \\ T & T & T & T & T & F & F & T \\ \hline \end{array}$$

$$\begin{array}{|cccc||cccc|} \hline p & \and & \neg & q & \neg & (p & \implies & q) \\ \hline F & F & T & F & F & F & T & F \\ F & F & F & T & F & F & T & T \\ T & T & T & F & T & T & F & F \\ T & F & F & T & F & T & T & T \\ \hline \end{array}$$