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Math Help - propositional logic

  1. #1
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    propositional logic

    hey friends,
    please help me out with this question! thanks lots!
    show that (p→q)∧(p→r) and p→(q∧r) is logically equivalent
    thanks in advance!



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  2. #2
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    Re: propositional logic

    Quote Originally Posted by llapkan View Post
    show that (p→q)∧(p→r) and p→(q∧r) is logically equivalent
    thanks in advan
    \begin{gathered}  (p \to q) \wedge (p \to r) \equiv  \hfill \\  (\neg p \vee q) \wedge (\neg p \vee r) \equiv  \hfill \\  \neg p \vee (q \wedge r) \equiv   p \to (q \wedge r) \hfill \\ \end{gathered}
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  3. #3
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    Re: propositional logic

    Another way to do this is to construct a "truth table". p, q, and r can each be "T" or "F" so there are 2^3= 8 cases.

    If p= q= r= T, then p\to q is true and p\to r is true so (p\to q)\wedge(\p\to r) is true. Also then [tex]q\wedge r[/itex] is true so p\to (q\wedge r) is true. Since they are both true (p\to q) \wedge (p\to r)= p\to (q \wedge r) in this case.

    If p= F, which includes 4 of the 8 cases, each of p\to q, p\to r, and p\to(q\wedge r) is true because " F\to x" is true no matter what x is.

    Now show they are the same in the other 3 cases.
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