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.
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.
No
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.
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.
Also, Not (A XOR B)