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Find the smallest positive integer solution (non-zero) for the following inequalities:

$a_1<c_1,a_2<b_1,d_1<a_3,c_2<a_4,c_3<b_2,c_4<b_3,b_4<d_2$

Under the following constraints

$a_1 \neq a_2 \neq a_3 \neq a_4,b_1 \neq b_2 \neq b_3 \neq b_4,c_1 \neq c_2 \neq c_3 \neq c_4, d_1 \neq d_2 $

Is there any method to solve this type of inequalities.

usr1092803564
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  • Welcome to MathSE! You are more likely to get a good answer to your question if you follow a few guidelines. In particular, what have you tried so far, and just where are you stuck? This is not a homework-answering site: we want to see that you have put significant work into the problem. Also, the question must be clear. In each multiple inequality, do you mean that all four numbers are distinct from each other or just that the two values on each side of each inequality are distinct? (E.g. is $a_1=a_3$ allowed?) – Rory Daulton Feb 28 '16 at 18:22
  • Thanks for important guidelines. $a_i \neq a_j , b_i \neq b_j , c_i \neq c_j , d_i \neq d_j $ for any $i$ and $j$. – usr1092803564 Feb 28 '16 at 18:29
  • Why negative vote? – usr1092803564 Feb 28 '16 at 18:31
  • I did not give you a downvote but I did vote to close this question. My most serious concern was the ambiguity in the multiple inequalities, and you have now addressed that concern. However, you still have not shown any work of your own. If you give us your own thoughts, what have you tried so far, and/or just where your are stuck I will remove that close vote. We hope you continue at this site, with good quality questions and answers. – Rory Daulton Feb 28 '16 at 18:38

1 Answers1

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I obtain a solution $$a_i=(1,2,3,4)$$ $$b_i=(3,4,2,1)$$ $$c_i=(4,2,3,1)$$ $$d_i=(1,2)$$ Let me tell you what I did, First of all, put everything as $1$, and then focus on the first constraint, you get $$a_i=(1,1,2,2)$$ $$b_i=(2,2,2,1)$$ $$c_i=(2,1,1,1)$$ $$d_i=(1,2)$$ (Without using the second constraint)

Now focus on the variables that are always lesser than some others, eg $a_1$, And put all of them as $1$. Next start focusing on the other variables and use that if $a>b$ you had $b=1$, so put $a=2$, and after a while you will get a solution.

Nikunj
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