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I have a question in a book I am solving(Discrete Structures by Kolman, Busby & Ross). I am unable to make sense from the question. It is stated below, Show that k is odd is a necessary and sufficient condition for k^3 to be odd.

Now what I extracted out of the question was, As k is odd is necessary as well as sufficient condition for k^3 then they must be the same logic I must say that they are logically equivalent. Is this what the question demands? Please help me understand the question.


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The question asks you to prove: $k^3$ is not divisible by $2$ if and only if $k$ is not divisible by $2$. – Asaf Karagila Sep 29 '11 at 21:34
"P is a necessary and sufficient condition for Q" is the same as "P is equivalent to Q", "P if and only if Q" ("P $\iff$ Q"). All of them mean "(If P, then Q) AND (If Q, then P)". – Srivatsan Sep 29 '11 at 21:36
I always tell... – The Chaz 2.0 Sep 29 '11 at 22:40
Like Chaz, I don't understand the downvote here... – J. M. Sep 30 '11 at 0:51
up vote 8 down vote accepted

To say that condition $P$ is necessary for condition $Q$ is to say that you cannot have $Q$ without having $P$ as well. That is to say: $$Q\rightarrow P$$

On the other hand, to say that a condition $P$ is sufficient for the condition $Q$ is to say that if you have $P$ then you surely have $Q$. Formally: $$P\rightarrow Q$$

Thus, to say that a condition is necessary and sufficient is to say that $P$ is sufficient for $Q$ and it is also necessary for it. Therefore this is to say that they are equivalent. $$P\leftrightarrow Q$$

The question which baffles you asks to show that $k$ is odd implies $k^3$ is odd, as well $k^3$ is odd implies $k$ is odd.

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$A$ is a necessary condition for $B$ means
i) $B\implies A$
ii) If $B$ is true then $A$ is true
iii) If $A$ is false then $B$ is false (contrapositive argument)

On the other hand, $A$ is a sufficient condition for $B$ means
i) $A\implies B$
ii) If $A$ is true then $B$ is true
iii) If $B$ is false then $A$ is false (contrapositive argument)

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