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Math Help - Question about verifying equivalence

  1. #1
    Junior Member
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    Question about verifying equivalence

    Can someone please explain how to verify/prove that:
    ((P => R1) ^ (P => R2)) <=> (P => (R1 ^ R2))
    and
    ((R1 => P) ^ (R2 => P)) <=> ((R1 v R2) => P)
    where "^" means "and", "=>" means "implies, and "<=>" means "equivalent to".
    They're obviously equivalent from glance, but I just don't know how to mathematically verify them, ie, manipulate left and right sides so that they're identical.

    Thanks.
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  2. #2
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    Recall that \left( {A \Rightarrow B} \right) \equiv \left( {\neg A \vee B} \right).
    So \left( {R_1  \Rightarrow P} \right) \wedge \left( {R_2  \Rightarrow P} \right) \equiv \left( {\neg R_1  \vee P} \right) \wedge \left( {\neg R_2  \vee P} \right) \equiv \left( {\neg R_1  \wedge \neg R_2 } \right) \vee P
    But \left( {\neg R_1  \wedge \neg R_2 } \right) \vee P \equiv \neg \left( {R_1  \vee R_2 } \right) \vee P \equiv \left( {R_1  \vee R_2 } \right) \Rightarrow P.

    Now you show the others.
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  3. #3
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    Thank you!
    Worked perfectly (=
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