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The true statements are: phi 1 OR phi 2 is satisfiable Here's why: Satisfiable Formula: A satisfiable formula is true under at least one interpretation
The true statements are:
phi OR phi is satisfiable
Here's why:
Satisfiable Formula: A satisfiable formula is true under at least one interpretation of its variables.
Implications of phi and phi being satisfiable:
phi OR phi must be satisfiable: If phi is true for a particular interpretation, then phi OR phi is also true for that interpretation, regardless of phis truth value. Similarly, if phi is true, phi OR phi is true. Therefore, at least one of the possible interpretations will make phi OR phi true, making it satisfiable.
Other statements are not necessarily true:
NOT phi is not necessarily unsatisfiable: It could also be satisfiable. If phi is true for some interpretations, then NOT phi is false for those interpretations, but it could be true for other interpretations where phi is false.
NOT phi is not necessarily valid: It could be valid or satisfiable, depending on the specific formula phi
phi AND phi is not necessarily satisfiable: It's only satisfiable if there exists an interpretation where both phi and phi are true simultaneously. This might not always be the case, even if both formulas are individually satisfiable.
Therefore, the only statement that is always true given the satisfiability of phi and phi is:
phi OR phi is satisfiable
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