Question: I'm simulating a auction game with sealed single bid, where each of the $n$ players has winning probability $p\_i,i=1,...,n$, and their bids $b\_i$ have to be calculated to meet the $p\_i$. Supposing that the value $v\_i$ of each player is uniformly distributed in the interval $[0,1)$ and $b\_i$ is a function of $v\_i$ (as usual), how could I calculate their bids to match the $p\_i$?

Answer: 
DeMorgan's laws:

1. $\sim (x \vee y) = (\sim x ) \wedge (\sim y)$
2. $\sim (x \wedge y) = (\sim x ) \vee (\sim y)$

In some notations where the overline indicates the complement, the rule is "break the line, change the sign":

1. $\overline{x+y} = \overline{x} \ \overline{y}$
2. $\overline{xy} = \overline{x} + \overline{y}$

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By only manipulating the right hand side, we have
$$( \sim q \ \lor \sim p) \wedge (\sim r \ \lor \sim p) \\ 
= [\sim (q\wedge p)] \wedge [\sim (r \land p)] \\ 
= \sim [(q\wedge p) \lor (r \land p)] \\
= \sim [(p \land q) \lor (p \land r)] \\
= \sim [p \land (q \lor r)] $$

which is contradictory (false in all possible worlds) to the original left hand side.
The left hand statement is equivalent to $$(p\land q)\lor (p\land r)$$

Its negation is precisely the right hand statement. So this is indeed a self-contradictory proposition.