if vs. iff

shahar

shahar

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What are the differences between the two?
Can anybody give an special example (and rarely known, if possible) for that?
 
shahar said:
What are the differences between the two?
Can anybody give an special example (and rarely known, if possible) for that?
Click to expand...
The hypothetical \(\text{ P implies Q }\) has three logical forms.
\(\text{If P then Q. }\)
\(\text{ P only if Q }\)
\(\text{ not P or Q }\)
Thus \(\text{ A if and only if B }\) means \(\text{ A implies B AND B implies A. }\)
 
shahar said:
What are the differences between the two?
Can anybody give an special example (and rarely known, if possible) for that?
Click to expand...
I have no idea why you want a rare example. How do you learn from that?

If n is evenly divisible by 10, then, necessarily, n is evenly divisible by 2.

But it is false that if n is evenly divisible by 2, then, necessarily, n is evenly divisible by 10.

So the first statement is an example of an "if" statement. It is necessarily true in only one direction.

If n is evenly divisible by 10, then, necessarily, n is evenly divisible by both 2 and 5.

If n is evenly divisible by both 2 and 5, then, necessarily, n is evenly divisible by 10.

Those are examples of "iff" statements; they are necessarily true in either direction.
 
"If x is divisible by 6 then x is divisible by 3", which is the same as
"x is divisible by 3 if it is divisible by 6", is true.

It is NOT true that "x is divisible by 3 if and only if x is divisible by 6" because that
would mean not only "x is divisible by 3 if it is divisible by 6" but also "x is divisible by 3 only if x divisible by 6" which is not true because 3 itself is divisible by 3 without being divisible by 6.
 
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