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.
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Recall that p = 1100 and q = 1010.
1) 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.
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