Some authors do not consider the empty set to be irreducible ( even though it vacuously satisfies the above conditions ).
12.
It begins vacuously ( its first two sentences are :'It has been said that America is great because America is good.
13.
A statement S is " vacuously true " if it resembles the statement P \ Rightarrow Q, where P is known to be false.
14.
If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).
15.
In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction.
16.
Ionesco was suggesting that by vacuously repeating cliches instead of meaningful communication, his characters had lost their ability to think critically and were thus already partly rhinoceros.
17.
The independent variable " x " does not appear on the right side of the function expression and so its value is " vacuously substituted ".
18.
I have not forgotten that Wikipedia can back up its jibes only with empty, inflammatory rhetoric, the very thing it vacuously accuses its opponents of using.
19.
A posetal category vacuously has all equalizers and coequalizers, whence it is ( finitely ) complete if and only if it has all ( finite ) products, and dually for cocompleteness.
20.
Cousin's theorem states that for every gauge \ delta, such a \ delta-fine partition " P " does exist, so this condition cannot be satisfied vacuously.
