What does this mean? Whats the difference between IFF and IF?

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Discussion Overview

The discussion centers on the distinction between "if" and "if and only if" (IFF) in logical implications. Participants explore the definitions and implications of these terms in the context of logical reasoning and proofs.

Discussion Character

  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants note that logical implication A -> B can be expressed as "if A then B."
  • Others clarify that A -> B and B -> A can be expressed as "iff A then B."
  • One participant states that A <-> B indicates that A and B have the same truth value, meaning both are true or both are false.
  • Another participant suggests that proving an IFF statement requires two proofs: one for A -> B and another for B -> A, with the possibility that one direction may be easier than the other.

Areas of Agreement / Disagreement

Participants appear to agree on the definitions and implications of "if" and "iff," but there is no explicit consensus on the ease of proving each direction in IFF statements.

Contextual Notes

Some assumptions about the audience's familiarity with logical notation and implications may be present, but these are not explicitly stated.

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What does this mean? Whats the difference between IFF and IF?
 
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BloodyFrozen said:
What does this mean? Whats the difference between IFF and IF?
Logical implication A -> B can be expressed as if A then B.

A -> B and B -> A can be expressed as iff A then B.
 


mathman said:
Logical implication A -> B can be expressed as if A then B.

A -> B and B -> A can be expressed as iff A then B.

Oh, I see now.

A<->B is iff

and A->B is if
 


BloodyFrozen said:
Oh, I see now.

A<->B is iff

and A->B is if

Another way to think about it is that A \Leftrightarrow B means that A and B have the same truth value. Either both are true or both are false.

If you are asked to do a proof of an IFF statement, you should break it down into two proofs: one of A \Rightarrow B and the other of B \Rightarrow A. Typically one direction will be easier than the other.
 

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