# How does the universal set apply to the truth set?

If the truth set is identical to it's universal set, then the statement is logically true statement and it's called a tautology.

Can someone please explain the universal set to me in this context? I thought the universal set has to be defined for each problem, so how is it defined, for, say, $$\neg (p\wedge q)\iff (\neg p) \vee (\neg q)$$

I'm not taking a class, just interested in learning more about math. Thanks 😊

• Hi, I edited a typo (I think!) and made the symbols more pretty. You can learn how to do that yourself here: math.meta.stackexchange.com/questions/5020/… . Please check if the symbols are correct now :) Aug 3 '20 at 5:32
• It would help if you gave a definition of truth set and universal set, I don't think it is common terminology. Aug 3 '20 at 9:34
• The notions of truth sets or universal sets rarely if ever come up in mathematics. There are only a few basic rules of logic that you need to remember, e.g. if A implies B then not B implies not A (the contrapositive rule). To learn them, you might Google "software to learn the basic methods of proof." Aug 3 '20 at 18:53
• @Calvin, that certainly is attractive! I will take some time to learn this mark-up 😁 thanks for the great resource Aug 4 '20 at 5:02

The terms "truth set" and "universal set" aren't really common, but from context it sounds like they mean the following:

Fixing a propositional language $$P$$, the universal set associated to $$P$$ is the set of all valuations of propositional formulas built from $$P$$ (or essentially equivalently, the set of all maps $$P\rightarrow\{True, False\}$$), and for a formula $$\varphi$$ built from $$P$$ the truth set of $$\varphi$$ is the subset of the universal set consisting of all valuations making $$\varphi$$ true.

Now you are correct that $$\varphi$$ alone can't tell us what $$P$$ is - e.g. if $$\varphi$$ is $$p\rightarrow q$$, then $$\{p,q\}$$ and $$\{p,q,r\}$$ would each make sense. However, $$\varphi$$ does have a minimal language associated to it, namely the set of all propositional atoms which actually occur in $$\varphi$$, so we often default to that language.

Moreover, in this context - and indeed in many contexts - the choice of language doesn't matter:

Suppose $$P_1,P_2$$ are two languages each of which contains every propositional atom occurring in $$\varphi$$. Then $$\varphi$$ is a tautology in the sense of $$P_1$$ iff $$\varphi$$ is a tautology in the sense of $$P_2$$.

This is a good exercise: the key point is that whether or not a valuation $$v$$ makes a sentence $$\varphi$$ true is determined entirely by the restriction of $$v$$ to the propositional atoms actually occurring in $$\varphi$$. So we don't have to worry about the choice of language issue here.

That said, sometimes the choice of language does matter. For example, in the context of first-order logic it is important to specify the language when talking about the decidability or completeness of a theory: Presburger arithmetic, for example, is complete and decidable as a $$\{+\}$$-language but not as a $$\{+,\cdot\}$$-language.