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.
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.
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.
If B then A is the same as if X then Y is the same as if A then B. They are saying it’s the same as the OP. Changing the letters around doesn’t change the meaning since the letters are just placeholders.
Now if you said If A then B AND If B then A as one it wouldn’t be the same because A and B would have to keep the same meaning.
Yeah if vs iff can be confusing at first. Trying to understand it with normal grammar doesn’t work right. It’s a lot more helpful to grok the symbols and so the truth tables by hand to see how they fit together
Is “If B then A” equal to “B if and only if A”?
Also no.
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.
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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.
No. They are effectively the same statement.
(A <=> B ) = (A=>B AND B=> A)
Wait. If they are effectively the same statement, wouldn’t that mean they ARE equal?
If B then A is the same as if X then Y is the same as if A then B. They are saying it’s the same as the OP. Changing the letters around doesn’t change the meaning since the letters are just placeholders.
Now if you said If A then B AND If B then A as one it wouldn’t be the same because A and B would have to keep the same meaning.
But they switched the order in only the first half of the statement. I don’t know if everyone commenting caught that.
Is “If B then A” equal to “B if and only if A”?
This IS different from the original question.
I mean it is the definition of “if and only if”. And by commutativity we also know that A iff B is equal to B iff A
Oh yeah. I was very confused as to what that meant, but I learned something today.
Yeah if vs iff can be confusing at first. Trying to understand it with normal grammar doesn’t work right. It’s a lot more helpful to grok the symbols and so the truth tables by hand to see how they fit together
Also, Not (A XOR B)