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16 November, 02:06

For each of these compound propositions, use the conditional-disjunction equivalence (Example 3) to find an equivalent compound proposition that does not involve conditionals.

(a) ~p→ ~q

(b) (p v q) → ~p

(c) (p→ ~q) → (~p →q)

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  1. 16 November, 02:28
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    (a) p v ~q

    (b) ~ (p v q) v ~p

    (c) ~ (~p v ~q) v (p v q)

    Step-by-step explanation:

    The conditional-disjunction equivalence is:

    P→Q ⇔ ~P v Q

    To find an equivalent compound proposition without the conditionals (without the "→") you have to apply the previous equivalence and simplify if possible.

    a) ~p→~q

    In this case, P = ~p and Q = ~q

    Applying the equivalence:

    ~ (~p) v ~q

    p v ~q

    b) (p v q) → ~p

    In this case P = (p v q) and Q = (~p)

    Applying the equivalence:

    ~ (p v q) v ~p

    c) (p→~q) → (~p→q)

    In this case, you have to apply the conditional-disjunction equivalence for every conditional in the compound proposition.

    First, let P = (p→~q) and Q = (~p→q)

    ~ (p→~q) v (~p→q) (1)

    Now, you have to find an equivalent compound proposition for both (p→~q) and (~p→q)

    For (p→~q):

    Let P = p and Q=~q

    ~p v ~q

    For (~p→q)

    Let P = ~p and Q = q

    ~ (~p) v q

    p v q

    Then the expression (1) is:

    ~ (~p v ~q) v (p v q)
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