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

Replies

• 

  simplycoder

  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
• 

  rahul69

  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...