Math Is Fun Forum

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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

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

gio_osh

Member

Registered: 2021-03-23

Posts: 2

Re: disproof of an existential proposition

(1)  Use induction to show 2^n > 3n 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.

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

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

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