Determine whether each of these compoundpropositions is satisfiable. a) (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧ (¬p ∨¬q∨¬s)∧(p∨q∨¬s) b) (¬p∨¬q∨r)∧(¬p∨q∨¬s)∧(p∨¬q∨¬s)∧ (¬p ∨¬r ∨¬s)∧(p∨q∨¬r)∧(p∨¬r∨¬s) c) (p∨q∨r)∧(p∨¬q∨¬s)∧(q∨¬r∨s) ∧(¬p∨ r ∨s) ∧(¬p∨q∨¬s)∧(p∨¬q∨¬r) ∧(¬p∨ ¬q ∨s)∧(¬p∨¬r∨¬s)

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Step-by-step explanation:

A compound proposition is satisfiable if there exists a combination of truth values of the component propositions that make the compound proposition true.

A compound conjunction p1∧p2∧p3∧...∧pn is true iff pi is true for all i.

A compound disjunction p1∨p2∨p3∨...∨pn is true iff pi is true for some i.

a) (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧(¬p∨¬q∨¬s)∧(p∨q∨¬s) is satisfiable.

Assume that p is true, q is true, r is false and s is false.

Since p is true, then (p∨q∨¬r), (p∨¬q∨¬s) and (p∨¬r∨¬s) are true.

Now, s is false, then ¬s is true. Then (¬p∨¬q∨¬s) and (p∨q∨¬s) are true.

Combine both statements to obtain that  (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧(¬p∨¬q∨¬s)∧(p∨q∨¬s) is true.

b) (¬p∨¬q∨r)∧(¬p∨q∨¬s)∧(p∨¬q∨¬s)∧(¬p∨¬r∨¬s)∧(p∨q∨¬r)∧(p∨¬r∨¬s) is satisfiable

Assume that p is false, q is false, r is false and s is false.

Since p is false, ¬p is true, then (¬p∨¬q∨r), (¬p∨q∨¬s) and (¬p∨¬r∨¬s) are true.

Since s is false, ¬s is true, then (p∨¬q∨¬s) and (p∨¬r∨¬s) are true.

Since s is false, ¬s is true, then (p∨q∨¬r) is true.

Combining all of these, the compound proposition is satisfiable (each component of the disjunction is true).

c) (p∨q∨r)∧(p∨¬q∨¬s)∧(q∨¬r∨s)∧(¬p∨r∨s)∧(¬p∨q∨¬s)∧(p∨¬q∨¬r)∧(¬p∨¬q ∨s)∧(¬p∨¬r∨¬s) is satisfiable.

Assume that p is true, q is true, s is true and r is false.

Since p is true, (p∨q∨r), (p∨¬q∨¬s) and (p∨¬q∨¬r) are true.

Since s is true, (q∨¬r∨s), (¬p∨r∨s) and (¬p∨¬q ∨s) are true.

Since ¬r is true, (p∨¬q∨¬r) and  (¬p∨¬r∨¬s) are true.

Since q is true, (¬p∨q∨¬s) is true.

Combining all of the above, the compound proposition is satisfiable.

Remark: there are unsatisfiable propositions, but proving that a proposition is unsatisfiable requires a general argument. For example, (p∧q)∧(¬p) is unsatisfiable, since it contains the contradiction (p∧¬p).

step-by-step explanation:

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