Determine if the following are tautologies contradiction or

Determine if the following are tautologies, contradiction or contingencies. You cannot use truth tables to justify your answers. Use either logical equivalences or some other means that does not use truth tables. (p^-q) doubleheadarrow (p rightarrow q) -(-(-a -b) rightarrow (z rightarrow (a b))) (-z -y) rightarrow -((z x (-x y))

Solution

a) p and negative q if true then p is true and q not true.

p true and q not true does not imply p implies q.

Since p and negative q can never imply p implies q.

So always false.

Contradiction

b) negative of negative of negative a or b is equivalent to

negative of (a and b)

again equivalent to ~a v ~b

Hence again contradiction.

c) Negative z and negative y on left side

Right side it is negative z or x or negative x or y

= ~(z v y)

= ~z ^ ~y

So Tautology