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My earlier question became too long, so succinctly:

Suppose $P(C)=0.2.$ Its complement is $0.8;$ i.e., $P(C)^\complement=0.8.$

But what does $P(¬C)$ mean? I think I am mixing up the terms 'complement' and 'negation'?

ryang
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hhh
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    If you are thinking of $C$ as a set, you should use the complement. If you are thinking of $C$ as a statement, you should use the negation. – Gerry Myerson Feb 26 '13 at 00:54
  • @GerryMyerson what about if I think them as both, can I use the notations interchangeable? – hhh Feb 26 '13 at 01:12
  • That depends on whether you are doing things for your own benefit, or trying to communicate them to someone else. In the first case, I don't think you can go wrong. In the second case, you have to be very careful not to confuse the other person/people involved. – Gerry Myerson Feb 26 '13 at 01:55

2 Answers2

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Informally, exactly the same thing is intended. We can think of events as sets (nowadays preferred) or as assertions (we got $4$ or more heads). The complement $A^c$ of an event $A$ is the set language version of "$A$ doesn't occur." The assertion version is to say that "not $A$" occurs, or, in symbols, that $\lnot A$ occurs.

A similar thing can happen with conjunction. In probability, it is usual to think of events as sets, so we ordinarily write $A\cap B$ for "$A$ and $B$." But some people still write $A\land B$, where $\land$ is the "and" of logic.

Thus, in your example, $\Pr(C^c)=\Pr(\lnot C)=0.8$.

André Nicolas
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    We have set-theory, probability-theory and quantifier-logic. When you mess them together, you get their mix -- is there some special name for this? Anyway +1 accepted, I think you answered the question but would like to know a bit more -- there must be something to connect the dots, thinking. – hhh Feb 26 '13 at 01:16
  • When we nowadays do probability theory formally, it is set-theoretically. There is the underlying set $\Omega$ of possible outcomes. Events are (sometimes with restrictions) subsets of $\Omega$. This is in line with the fairly general pattern of using set-theoretic language for everything. But that is not the way early probabilists thought, and it is an unnatural way to think of "the probability of rain today is $0.80$" (though it can be done). A more traditional view is to associate probabilities with certain sentences (I will be dealt a straight flush in the poker game. – André Nicolas Feb 26 '13 at 01:28
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    This enables us to assign probabilities to certain combinations of sentences, and that's where logical connectives come into the game. But it can all be done purely set-theoretically. – André Nicolas Feb 26 '13 at 01:29
  • Yes, it is freaking-cool -- I have seen once a guy doing probability with set-theory and turning logic into set-theory, it looks funny. You axiomatize everything. You can even turn simple arithmetics into set-theory where you redefine numbers and arithmetic operations into sets. I don't know whether there is any specific name for it, perhaps not. – hhh Feb 26 '13 at 01:37
  • Yes, in principle almost all of mathematics (excluding parts of Category Theory) can be done within formal set theory. The standard set theoretic basis for mathematics is Zermelo-Fraenkel set theory, usually with Axiom of Choice added (ZFC). Number theory, analysis, group theory, partial differential equations, combinatorics, whatever: it can all be developed in ZFC. *All objects mathematics deals with are then sets. – André Nicolas Feb 26 '13 at 01:48
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Suppose $P(C)=0.2.$ Its complement is $0.8,$ i.e., $P(C)^\complement=0.8.$ But what does $P(¬C)$ mean? I think I am mixing up the term complement and negation?

The negation of the sentence $$P(C)=0.2$$ is $$P(C)\ne0.2,$$ whereas the complement of the event (set) $$C$$ is $$C^\complement.$$

The negation of a tautology is a contradiction;

on the other hand, the complement of the set of tautologies contains contingent sentences as well as contradictions.

ryang
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