Answer to Question #264370 in Statistics and Probability for Liz

Question #264370

Use rules of inference and laws of logical equivalence to prove the following:

(p → q) ∧ (r → s) ∧ [t → ¬(q ∨ s)] ∧ t ⇒ (¬p ∧ ¬r)


1
Expert's answer
2021-11-12T08:02:38-0500

"p \u2192 q\\equiv \\neg p\\lor q"

"r \u2192 s\\equiv \\neg r\\lor s"

"t \u2192 \u21c1(q \u2228 s)\\equiv \\neg t \\lor \\neg(q \u2228 s)"

"( \\neg t \\lor \\neg(q \u2228 s))\\land t \\equiv \\neg(q \u2228 s)\\land t"


if t is true, then:

"\\neg(q \u2228 s)\\land t \\equiv \\neg(q \u2228 s) \\equiv \\neg q \\land \\neg s"


then we get Destructive Dilemma that is Tautology:

"(p \u2192 q) \u2227 (r \u2192 s) \u2227 (\\neg q \\land \\neg s) \\implies (\u00acp \u2227 \u00acr)"


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