In the following four exercises use Boole to construct truth

In the following four exercises, use Boole to construct truth tables and indicate whether the sentence is tt-possible and whether it is a tautology. Remember how you should treat long conjunctions and disjunctions. ~[(~A U B) ^ ~(C ^ D)]

Solution

A B C D ~A AUB (~AUB) C^D ~(C^D) [(~AUB)^ ~(C^D)] ~ [(~AUB)^ ~(C^D)]

0 0 0 0 1 0 1 0 1 1 0

0 0 0 1 1 0 1 0 1 1 0

0 0 1 0 1 0 1 0 1 1 0

0 0 1 1 1 0 1 1 0 0 1

0 1 0 0 1 1 0 0 1 0 1

0 1 0 1 1 1 0 0 1 0 1

0 1 1 0 1 1 0 0 1 0 1

0 1 1 1 1 1 0 1 0 0 1

1 0 0 0 0 1 0 0 1 0 1

A expession is said to be tautaulogy if it is true for all cases. In this example i have checked till number 8 ,we have observed that for number 1,2,3 the expression is 0 or false.So it is not tautology