SYMBOLIC LOGIC!! Constructing a truth table?

Question

by constructing a truth table, determine whether or not the following sentence is tautological, contradictory, or contingent. Include all columns and rows of the truth table.

[(Q>P) V (~Q>R)] V ~(P@R)

> is implication, commonly →.

"<" means equivalence, commonly ≡,

@, is and

V, is or

Answer 1

I won't do it for you but I'll give you a few clues as to how to do it yourself.
Since you have 3 variables, you'll need 2^3 or 8 rows. Begin by writing P, Q, and R at the top of the first three columns, and all possible values T and F under each variable. For P write 4 T's and 4 F's. For Q alternate by 2's, and for R alternatie by 1's.
Then write the statement across the top of the truth table and, working from inside the parenthesis first, write the truth value of each component. Work from the innermost to outermost parenthesis, and left to right. When you get to the last column you can read the result.

Answer 2

P Q R (Q → P) (~Q → R) ~(P ∩ R)
0 0 0 1 0 1
0 0 1 1 1 1
0 1 0 0 1 1
0 1 1 0 0 1
1 0 0 0 0 1
1 0 1 0 1 0
1 1 0 1 1 1
1 1 1 1 0 0

Answer 3

Oh come on - there's only 3 variables and thus only 8 rows in the truth table - can you really not do this yourself? At least make an attempt and publish it here and ask people to check it over.

Answer 4

If P is true you have
[(Q → t) v (~Q → R)] v ~(t & R)
[ t v (stuff)] v ~R
[ t ] v ~R
t

If P is false you have
[(Q → f) v (~Q → R)] v ~(f & R)
[ stuff ] v ~(f)
[ stuff ] v t
t

So without a truth table in all its tedious glory, you know you have a tautology.