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I have seen various theorems, where in statement "if and only if" is written and in the proof they call one statement as necessary and other as sufficient. I often get confused regarding the relationship of "if" and "only if" with "necessity" and "sufficiency or converse". My question is, "A holds if and only if B holds".In this statement which part is necessary and which is converse between the following two: 1)suppose A holds, then B follows. 2)Suppose B holds, then A follows.

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  • Do this and this help clarify your thoughts? – Ken Hung Jul 13 '21 at 04:36
  • Suppose that P being true implies that Q is true, i.e., $P\implies Q.$

    This means precisely that there is no way for P to be true and Q false.

    In other words: P is true only if Q is true.

    Similarly, $P\rightarrow Q$ is read as $“P$ only if $Q.”$

     – ryang Jul 13 '21 at 05:59

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