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#1 2021-03-23 18:39:02

gio_osh
Member
Registered: 2021-03-23
Posts: 2

disproof of an existential proposition

I would like to disproof:

.

So far I've got:

     and     
.
But I am unsure on how to proceed next. Any help, answer, or suggestion would be appreciated.

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#2 2021-03-23 20:18:01

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: disproof of an existential proposition

hi gio_osh

Welcome to the forum.

I think you could do this by proving that 3x < 2^x for all x > 4

Two induction proofs needed.

(1)  Use induction to show 2^n > 3 for n > 4

(2) (a) Show 3 times 5 > 2^5

(b) Assume 3n < 2^n (n > 5) : add 3 to each side and in a couple of steps you have 3(n+1) < 2^ (n+1)

That should do it.

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 smile

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#3 2021-03-24 16:30:29

gio_osh
Member
Registered: 2021-03-23
Posts: 2

Re: disproof of an existential proposition

Bob wrote:

(1)  Use induction to show 2^n > 3n for n > 4

Bob wrote:

(2) (a) Show 3 times 5 < 2^5

Bob wrote:

(b) Assume 3n < 2^n (n > 5) : add 3 to each side and in a couple of steps you have 3(n+1) < 2^ (n+1)

Bob wrote:

That should do it.

Thanks for your answer. Can you tell me if I missed something please.

Last edited by gio_osh (2021-03-24 16:32:57)

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#4 2021-03-24 21:05:15

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: disproof of an existential proposition

hi gio_osh

That's what I had in mind.  When you use induction you have to make it clear you are assuming the result is true for 'n' and using that to show it is therefore true for 'n+1'.

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 smile

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