What is the reason for the logical equivalence between the predicates?

radha gogia

radha gogia

@radha-BTDzli • Oct 26, 2024
Just little confusion in the logical equivalence of the 2 below logical expressions:
1. (p->r) AND(q->r)= (p OR q)->r
2. (p->r) OR (q->r)=(p AND q)->r
In these expressions ,I am unable to get out the reason why while doing AND operation ,we get OR of the two propositions p and q,so plz explain the reason for both expressions.
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  • simplycoder

    simplycoder

    @simplycoder-NsBEdD • Jan 9, 2015

    I would explain the first expression, you can think on for the second one on similar lines.
    1. (p->r) AND(q->r)= (p OR q)->r
    I hope you are convinced that the above expression is logically correct, if not then draw a truth table and check the same.

    We can always treat a->b as (~a) OR b. --------(1)
    Hence the LHS of first expression can written as
    ((~p) OR r)AND((~q) OR r))------(2)
    Therefore LHS now can be rewritten as ((~p)AND(~q)) OR r -----(3)

    ~p AND ~q is logically equivalent to ~(p OR q)
    Therefore(3) can be rewritten as
    ~(p OR q) OR r
    and by (1) is equivalent to
    (p OR q)->r


    Take care
    -SC
    0
  • rahul69

    rahul69

    @rahul69-97fAOs • Jan 10, 2015

    radha gogia
    Just little confusion in the logical equivalence of the 2 below logical expressions:
    1. (p->r) AND(q->r)= (p OR q)->r
    2. (p->r) OR (q->r)=(p AND q)->r
    In these expressions ,I am unable to get out the reason why while doing AND operation ,we get OR of the two propositions p and q,so plz explain the reason for both expressions.
    First you should study De Morgan's Law, then you will be able to understand this...
    0