# Birational Geometry of Foliations by Marco Brunella

By Marco Brunella

The textual content offers the birational type of holomorphic foliations of surfaces. It discusses at size the speculation built through L.G. Mendes, M. McQuillan and the writer to review foliations of surfaces within the spirit of the type of advanced algebraic surfaces.

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**Sample text**

Under what conditions is an implication true? false? Let’s begin with an example you are all familiar with. ” Under what conditions would he feel that we had lied? ” The conclusion, 2. ” Well, if our son cleans his room and we let him go to Henry’s, everybody is happy. That implication should be true. So if P is true and Q is true, the whole statement should be true. Also, it should be as clear to you as it will be to our son, that if he cleans his room and we do not let him go to Henry’s, we lied.

What does it mean for a number x to be odd? It means that there is an integer n such that x 2n + 1. So we are assuming that x2 2n + 1 for some integer n and trying to show x 2m + 1 for some integer m. It’s hard to see where to go from here, we think. Remember that P´olya suggests restating the problem, so let’s try that. ” Then we see that we wish to prove that P → Q is true. ” We can do better than that, since an integer is either odd or even. So we can show that “If x is even, then x2 is even” and that will be equivalent.

If she eats bread or yogurt, she also eats cereal. She never eats both cereal and yogurt. She always eats bread or cereal. Can you say what Matilda eats on Monday? If so, what does she eat? 7. Consider the following statement. If the coat is green, then the moon is full or the cow jumps over it. (a) This unusual statement is composed of several substatements. Identify each substatement, give it a letter, and write down the original statement using these letters and logical connectives. (b) Using the symbols introduced in (a), ﬁnd the contrapositive of the original statement.