Friday, July 9, 2010

Practice problems for logic and tautology

The text editor for this blog doesn't have any arrow symbol, either for mapping or for implication. Instead, the mapping arrow will be written as -> and the implication arrow will be =>.

For all these problems, let p = 1100 and q = 1010. Determine if the following logical statements are tautologies (all 1's), contradictions (all 0's) or conditional (some 1's and some 0's).

Problem 1.
(p v q) => ~p
Problem 2.
p v (q => ~p)

Problem 3.
(p ^ q) => ~p

Problem 4.
(p ^ ~p) => q

1 comment:

Prof. Hubbard said...

Problem 1.
(p v q) => ~p

1100 v 1010 = 1110

Change implication to ~premise v conclusion.
~1110 = 0001

0001 v 0011 = 0011.

This statement is conditional.

(Thanks to Phillip Barringer for the correction.)

Problem 2.
p v (q => ~p)

q => ~p becomes ~q v ~p

0101 v 0011 = 0111

1100 v 0111 = 1111.

This statement is a tautology.

(This one also needed to be corrected.)

Problem 3.
(p ^ q) => ~p

p ^ q = 1100 ^ 1010 = 1000

~1000 = 0111

0111 v 0011 = 0111.

This statement is conditional.

Problem 4.
(p ^ ~p) => q

p ^ ~p = 1100 ^ 0011 = 0000

~0000 = 1111

1111 v 1010 = 1111.

This statement is a tautology. (Any implication where the premise is a contradiction will be a tautology.)