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#1 2017-06-11 10:49:30

chrislav
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Registered: 2017-06-10
Posts: 5

high school inequalities

Prove ;

1) a>=b and b>=c => a>=c

2) a>=b and b>=a => a=b

Last edited by chrislav (2017-06-11 17:51:03)

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#2 2017-06-11 18:58:26

Bob
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Registered: 2010-06-20
Posts: 10,363

Re: high school inequalities

hi chrislav

Welcome to the forum.

Some people, looking at these, might say 'isn't it obvious?'  So I'm guessing this is a proof from first principles analysis.  Tp start you need to look at the definition of >=

If you post this back I'll see if I can help.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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#3 2017-06-11 20:27:53

chrislav
Member
Registered: 2017-06-10
Posts: 5

Re: high school inequalities

bob bundy wrote:

hi chrislav

Welcome to the forum.

Some people, looking at these, might say 'isn't it obvious?'  So I'm guessing this is a proof from first principles analysis.  Tp start you need to look at the definition of >=

If you post this back I'll see if I can help.

Bob

Thanks Bob ,you are right.

So the axioms and the  definition needed for the above proofs are:

Axioms:

1) the trichotomy law for ">"

2) a>b and b>c => a>c

3) a>b => a+c>b+c

4) a>b and c>0 => ac>bc

Definition : a>=b <=> a>b or a=b

By the way isnt there any LaTex we can use??

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#4 2017-06-11 21:27:29

Bob
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Registered: 2010-06-20
Posts: 10,363

Re: high school inequalities

hi chrislav

Thanks for the quick reply.  Yes, you can use LaTex.  Look here: http://www.mathisfunforum.com/viewtopic.php?id=4397

That thread goes on for a long time but you'll read enough in the first few posts to get started.

Definition : a>=b <=> a>b or a=b

I think that definition is the place to start.  If you consider separate cases, eg a>b AND b=c, you can show the required result for each case.

Hope that helps

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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#5 2017-06-12 09:41:15

chrislav
Member
Registered: 2017-06-10
Posts: 5

Re: high school inequalities

bob bundy wrote:

hi chrislav

Thanks for the quick reply.  Yes, you can use LaTex.  Look here: http://www.mathisfunforum.com/viewtopic.php?id=4397

That thread goes on for a long time but you'll read enough in the first few posts to get started.

Definition : a>=b <=> a>b or a=b

I think that definition is the place to start.  If you consider separate cases, eg a>b AND b=c, you can show the required result for each case.

Hope that helps

Bob

O.K .Let us start:

using the definition

Which is equal to:

But if this is correct which is the rule in mathematics we use to get the above ??

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#6 2017-06-12 21:09:49

Bob
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Registered: 2010-06-20
Posts: 10,363

Re: high school inequalities

hi chrislav

That wasn't quite what I meant.  There are 4 cases:

case 1.  a > b AND b > c
case 2.  a > b AND b = c
case 3.  a = b AND b > c
case 4.  a = b AND b = c

If you are able to prove the required result for each of the above, then you're done.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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#7 2017-06-13 03:46:35

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: high school inequalities

Transitivity should be a part of the axioms of a partial order.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#8 2017-06-13 09:51:03

chrislav
Member
Registered: 2017-06-10
Posts: 5

Re: high school inequalities

bob bundy wrote:

hi chrislav

That wasn't quite what I meant.  There are 4 cases:

case 1.  a > b AND b > c
case 2.  a > b AND b = c
case 3.  a = b AND b > c
case 4.  a = b AND b = c

If you are able to prove the required result for each of the above, then you're done.

Bob

O.K BOB

But i would like to know,if possible, what ,mathematical rule dictates that there must be those 4 cases

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#9 2017-06-13 19:51:07

Bob
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Registered: 2010-06-20
Posts: 10,363

Re: high school inequalities

hi chrislav

It stems from the definition:

Definition : a>=b <=> a>b or a=b

This means that for a and b there are two cases.  And for b and c there are two cases.  Putting these together makes 2 x 2 cases.

I suppose this comes from basic logic theory, or from (probability) trees.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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#10 2017-06-15 04:24:58

chrislav
Member
Registered: 2017-06-10
Posts: 5

Re: high school inequalities

bob bundy wrote:

hi chrislav

I suppose this comes from basic logic theory, or from (probability) trees.

Bob

Thanks bob,sorry for being so inquisitive  .

But, i wonder, if we do not know the basic laws of logic and how are they ivolved in a mathemetical proof how can we be sure for the correctness of that proof??

Last edited by chrislav (2017-06-15 04:32:32)

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#11 2017-06-16 02:02:14

Bob
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Registered: 2010-06-20
Posts: 10,363

Re: high school inequalities

All I can say is they are fundamental to any mathematical theory so can be taken as "unstated" axioms.  The quoted definition uses "OR", so is already using a logical symbol, and the logical symbols "AND" & "IMPLIES" are also used in the axioms.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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