Answer to Question #231721 in Discrete Mathematics for rama

Question

Check whether the compound proposition

(p ∨￢q) ∧(q ∨￢r) ∧(r ∨￢p) ∧(p ∨q ∨r) ∧(￢p ∨￢q ∨￢r)

is satisfiable or not?

Answer 1

Using distribution property of  and  operators, the given proposition may be re-written as:

(p V((q) ∧(q ∨~r) ∧(~qV~s)∧(~rV~s)∧(qV~s)))∧(~pV~qV~s)

If we choose p  to be true, that is p=1, and q to be false, that is q=0, then  

(p V((q) ∧(q ∨~r) ∧(~qV~s)∧(~rV~s)∧(qV~s)))∧(~pV~qV~s)

equals 1 as p=1.

Also,  (~pV~qV~s) equals 1 as q=0  gives ~q=1.

Thus, we shall get the truth-value of the given proposition to be  for p=1  , q=0 and  and r and s may take either value  0 or 1 

Thus, the given proposition is satisfiable.