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Thanks. Could you possibly elaborate? Why are they not equal?
Substitute common sense terms. If I say "if it is an apple, it is a fruit", does it then follow that a thing is a fruit if and only if it is an apple? No. Lots of other things are fruit without being an apple.
Better read that one again.
If Apple then fruit. Is Apple ONLY if it's a fruit.
This one actually checks out.
If and only if is a biconditional. "b if and only if a" means "if b then a" AND "b only if a". B only if A here means "It is an apple only if is a fruit", in other words, "if it is a fruit, it could only be an apple." Which ain't right.
B -> A (if B, then A) (if apple, then fruit, correct)
B <-> A (B if and only if A) (if apple, then fruit, AND if fruit, then apple, incorrect).
Gotcha. I was reading it aloud: "It's an Apple if and only if it's a fruit." which isn't wrong, but I guess the technical definition of "If and only if" assumes more than the words imply.
"if and only if" is an unusual and sometimes confusing way to say it, but the words do directly imply the technical definition.
"it's an apple if and only if it's a fruit" literally means "it's an apple if it's a fruit" and "it's an apple only if it's a fruit". You already seem to understand the 2nd part, so no need to explain that.
The first part is a bit confusing because the words are in the reverse order compared to how people normally talk. "it's an apple if it's a fruit" means the same thing as "if it's a fruit, then it's an apple". Clearly "if it's a fruit, then it's an apple" is not a true statement, because there are plenty of other fruits apart from apples.
I agree, if and only if trips me up too and it doesn't fit perfectly into this logic formula. A thing is only an apple if and only if it is a fruit makes sense if you read it in a common sense way.