Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

Thanks.

share|improve this question
1  
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
2  
"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

2 Answers 2

up vote 7 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.

share|improve this answer

$A$ 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)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.