A little problem about mathmatical logic

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The discussion clarifies that the biconditional sign (P <=> Q) and the equivalence sign (P ≡ Q) are effectively the same in the context of mathematical logic, both indicating that two propositions have the same truth values. It emphasizes that when stating "P is defined as Q," it implies that P and Q are equivalent. The differences between the symbols are minor and primarily pertain to their specific contexts or nuances. For practical purposes, they can be treated interchangeably in most scenarios. Overall, understanding these symbols aids in grasping the relationships between propositions in logic.
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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.
 
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please help...
 
Yes, the "biconditional" and "equivalence" are the same thing. And if "A" is defined as being "B", then A and B are equivalent.
 
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.
 

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