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  • An example of why this is incorrrect.

    If a card is the ace of spades, it is black.

    A card is black if and only if it is the ace of spades.

    There are other conditions under which B (a card is black) can happen, so the second statement is not true.

    A conclusion that would be correct is "If a card is not black, it is not the ace of spades.". The condition is that if A is true B will also always be true, so if B is false we can be sure that A is false as well - i.e. "If not B, not A".

  • The first statement only tells you when B is true. It says nothing about when it is false. The second statement both tells you when B is true (if A) and when it is not (only if A). Therefore, the two statements cannot be equal.

58 comments