# Difference between revisions of "TADM2E 1.2"

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− | + | Due to the symmetric nature of the problem (swapping ''a'' and ''b'' does not change the problem), let | |

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− | + | ||

− | + | ||

− | -- | + | <math>min(a,b) = a</math> |

+ | |||

+ | We now have to show that there exist some real numbers for which the following holds true: | ||

+ | |||

+ | <math> ab<a </math> | ||

+ | <math>\implies a(b-1)<0 </math> | ||

+ | |||

+ | So, the product of two numbers (''a'' and ''b-1'') is negative. This means one of the numbers is negative and the other is positive. | ||

+ | |||

+ | ''Case I'' : | ||

+ | <math>a < 0 \and b-1>0</math> | ||

+ | <math>\implies a<0 \and b>1</math> | ||

+ | |||

+ | In this case, if one of the numbers is negative and the other is greater than 1 then <math> ab<min(a,b) </math> | ||

+ | |||

+ | ''Case II'' : | ||

+ | <math>a > 0 \and b-1<0</math> | ||

+ | <math>\implies a>0 \and b<1</math> | ||

+ | Since, <math> min(a,b) = a </math> | ||

+ | <math>\implies a<b</math> | ||

+ | <math>\implies 0<a<b<1 </math> | ||

+ | |||

+ | In this case, if both the numbers lie between 0 and 1 then <math> ab<min(a,b) </math> |

## Revision as of 18:24, 17 May 2016

Due to the symmetric nature of the problem (swapping *a* and *b* does not change the problem), let

$ min(a,b) = a $

We now have to show that there exist some real numbers for which the following holds true:

$ ab<a $ $ \implies a(b-1)<0 $

So, the product of two numbers (*a* and *b-1*) is negative. This means one of the numbers is negative and the other is positive.

*Case I* :
$ a < 0 \and b-1>0 $
$ \implies a<0 \and b>1 $

In this case, if one of the numbers is negative and the other is greater than 1 then $ ab<min(a,b) $

*Case II* :
$ a > 0 \and b-1<0 $
$ \implies a>0 \and b<1 $
Since, $ min(a,b) = a $
$ \implies a<b $
$ \implies 0<a<b<1 $

In this case, if both the numbers lie between 0 and 1 then $ ab<min(a,b) $