A little problem about mathmatical logic

[jeremy22511]

Homework Statement

1. Is there any difference between the following 2 signs?
<=> (for biconditional) and 三(the equivalence sign)
2. When we say 'P is defined as Q), do we mean P三Q?


Thanks

J

Homework Equations

The Attempt at a Solution

It seems that for 2 propositions, P & Q:
(i) P三Q when 'P and Q have the same kinds and numbers of components' and 'their truth values are equivalent'
(ii) P <=> Q is expressing the same thing as above.

 

[jeremy22511]

please help...

 

[HallsofIvy]

Yes, the "biconditional" and "equivalence" are the same thing. And if "A" is defined as being "B", then A and B are equivalent.

 

[Hurkyl]

For practical purposes, yes, they are effectively the same. The situation is similar to the four symbols $\rightarrow, \implies, \vdash, \models$.

Any difference between them is in the minor details -- so unless you're studying those, you can treat them as essentially the same.

From your description, it sounds like:

1. $P \Leftrightarrow Q$ is a proposition, formed by connecting the propositions P and Q with the binary symbol $\Leftrightarrow$? Or maybe your source is using $P \leftrightarrow Q$ for that...
2. $P \equiv Q$ is an assertion about the syntactic and semantic properties of the two propositions P and Q.