# Find an infinite set of formula, take any $2$ they are SAT, take any $3$ they are UNSAT [closed]

Find an infinite set of (propositional logic) formula, $$S$$, which satisfies:

• If you take any $$2$$ elements from $$S$$, they are satisfiable.
• If you take any $$3$$ elements from $$S$$, they are unsatisfiable.

As an illustration, a finite set of formula $$S' = \{a, b \land c, a \implies \neg b\}$$ satisfies the conditions as:

• Taking $$a$$ and $$b \land c$$, they are satisfiable with $$a = 1$$, $$b = 1$$, $$c = 1$$.
• Taking $$a$$ and $$a \implies \neg b$$, they are satisfiable with $$a = 1$$, $$b = 0$$, $$c = any$$.
• Taking $$b \land c$$ and $$a \implies \neg b$$, they are satisfiable with $$a = 0$$, $$b = 1$$, $$c = 1$$.
• Taking three of them, they are unsatisfiable.

Credit to my professor, to be honest I haven't solved this.

## closed as off-topic by JonMark Perry, ManyPinkHats, rhsquared, boboquack, QuintecDec 9 '18 at 22:02

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – JonMark Perry, ManyPinkHats, rhsquared, boboquack, Quintec
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• I am not too sure, but can anyone tell me whether this is off-topic, or seems to be a homework that should not be solved at puzzling.se? Thanks! – Omega Krypton Dec 9 '18 at 6:31
• Now I'm not sure if this is off-topic or not. But this is just a puzzle my professor gave on the class (as exercise), and everyone may try to solve it. – athin Dec 9 '18 at 6:51