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3

Sometimes we insist that the three literals in a 3SAT clause belong to different variables. This ensures, for example, that a random assignment satisfies a clause with probability exactly $7/8$. The translation $x \lor y \lor y$ doesn't satisfy this condition, but the other one does.


2

You just need to use $P_1$, $P_2$, $P_3$ and $\Leftrightarrow$ symbols, for example $P_2$ is a formula, $(P_1 \Leftrightarrow P_3) \Leftrightarrow P_2$ is a formula. As stated in the question, you must not use $\vee$, $\wedge$ or $\neg$. You need to find the number of non-equivalent formulae. For example, $P_1 \Leftrightarrow P_2$ is equivalent to $P_2\...


2

In the first code, the values taken by $(c, d)$ are in the set $\{(c, d)| 1 \leq d < c \leq x\}$. In the second code, the set is $\{(c,d)|1\leq c < d \leq x\}$. Since $c$ and $d$ play symmetrical roles in the boolean test, it is expected that you obtain the same result. Note that if $c$ and $d$ would not play symmetrical roles, the result could be ...


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