Online notes for Math 15, Math for Liberal Arts taught in the Fall 2016 semester at Laney College by Prof. Hubbard.
Monday, July 18, 2011
Tautology practice for Summer 2011.
When a logical statement is a tautology, the bitstring created when performing all the necessary operations is all 1s. In the following problems we will have two logical variables p and q, so p = 1100 and q = 1010. The AND operator is ^.
The OR operator is v.
The NOT operator is ~.
The IMPLIES operator should be an arrow pointing right, but since that single symbol is not an option in .html, I will use => instead. Remember that p => q can be changed to ~p v q.
Determine if each of these logical statements is a tautlogy or not.
1) p v (p => q)
2) p ^ (p => q)
3) (p v ~p) => q
4) (p ^ ~p) => q
Answers in the comments.
Recall that p = 1100 and q = 1010.
ReplyDelete1) p v (p => q)
p => q is the same as ~p v q.
~p v q =
~1100 v 1010 =
0011 v 1010 =
1011
if we take the answer and OR with p, we get 1100 v 1011 = 1111, so this statement is a tautology.
2) p ^ (p => q)
Again, p => q becomes ~p v q.
~p v q =
~1100 v 1010 =
0011 v 1010 =
1011
if we take the answer and AND with p, we get 1100 ^ 1011 = 1000, so this statement is NOT a tautology.
3) (p v ~p) => q
(p v ~p) =
1100 v 0011 =
1111
1111 => 1010 =
~1111 v 1010 =
0000 v 1010 =
1010
This statement is NOT a tautology.
4) (p ^ ~p) => q
(p ^ ~p) =
1100 ^ 0011 =
0000
0000 => 1010 =
~0000 v 1010 =
1111 v 1010 =
1111
This statement is a tautology.