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
Answers in the comments.
Problem 1.
ReplyDelete(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.)