Question

Does anyone know how to write a formal proof?

Answer 1

Yes, why?

Answer 2

In what context?

Answer 3

I have to write a proof of the following: If n is an integer, then 2n-5 is an odd integer

Answer 4

Here's an example:

\(\mathbf{Theorem}\quad\)If \(x=2\), then \(3+x=5\).
\(Proof.\) Let \(x=2\). Then \(3+x=3+2=5\). \(\square\)

Answer 5

To prove that, you need to know the definition of an odd number, that is, a number \(n\) is odd iff \(n=2k+1\), for some integer \(k\).

Answer 6

2n-5 = 2(n-2) -1

Answer 7

Right, my teacher gave us that but I don't know how I would use that

Answer 8

The book will walk me through this problem?

Answer 9

@luvspi 2n-5 = 2(n-2) -1

Any even number is a multiple of 2....

Answer 10

Ok estudier, is that step 1?

Answer 11

Actually, it is more of an invitation to think......:-)

Answer 12

But I don't know how to do formal proofs :'-(

Answer 13

Forget about formal proofs for a minute, your job is to convince your audience that your claim is correct.

So, your claim is "If n is an integer, then 2n-5 is an odd integer"

And I say, convince me.....

Answer 14

Here is one way to start:
you have the definition
x is odd if and only if x=2k+1, for some integer k.

given x= 2n-5, n an integer.
If you find an integer k that works, that would prove that x is odd right?

Answer 15

For any integer you plug into k, your going to get an odd number

Answer 16

His teacher (and me ) gave him 2n-5 = 2(n-2) -1

Answer 17

ok, just ran across campus back to my dorm to get on the coputer......i got that 2n-5=2n-5

Answer 18

for any inter you plug into 2n-5 you get an odd number( 2(0)-5 = -5.....2(1)-5=-3)...all odd

Answer 19

How do you know it works for all?

Answer 20

did your teacher give you a formal definition for an odd number?

Answer 21

Or an even number

Answer 22

so i have to come up with a universal statement that works for all integers?
she gave us a direct proof to an odd number: any integer n1 n=2k+1 for some integer k

Answer 23

even integer, n=2k for some integer k

Answer 24

So, 2n-5 = 2(n-2) -1

What can you say about the right hand side?

Answer 25

its equal to the left

Answer 26

Anything else?

Answer 27

What about the 2(n-2) part?

Answer 28

2 times a number minus 2.....you're subtraction two from the origional integer, thus keeping it either an odd or even number?

Answer 29

What if I say that (n-2) = k so that 2(n-2) = 2k?

Answer 30

okay, i get that

Answer 31

Up above, you said "even integer, n=2k for some integer k"

Answer 32

so the formula for an odd integer is the same minus 1?

Answer 33

Or + 1?

Answer 34

right

Answer 35

So now let's go back, your claim is

"If n is an integer, then 2n-5 is an odd integer" and you also said

2n-5 = 2(n-2) -1 and next....?

Answer 36

so basically its 2n- any odd number, and an odd number is 2k+1 so its 2(n-2)-2k+1?

Answer 37

Now you are jumping ahead to the end, better to do the inbetween steps.......

Answer 38

hmmm, inbetween?

Answer 39

"If n is an integer, then 2n-5 is an odd integer" and you also said

2n-5 = 2(n-2) -1 and next....?

Then you (we) said that since 2(n-2) is even, then.....?

Answer 40

then, errrrr i feel like luke, and you're yoda

Answer 41

:-)

Answer 42

So therefore, 2(n-2) -1 must be odd

Answer 43

but 2(n-2)- any odd number would be odd right?

Answer 44

-1, -3, -5, .....

Answer 45

Prove it....:-)

Answer 46

claiming
So therefore, 2(n-2) -1 must be odd
without proof is not kosher. You must show that this quantity fits the definition of odd
i.e there exists integer k s.t. 2k+1 = this number

Answer 47

"If n is an integer, then 2n-5 is an odd integer" and you also said

2n-5 = 2(n-2) -1 and next....?

Then you (we) said that since 2(n-2) is even, then.....?

So therefore, 2(n-2) -1 must be odd


Are we done?

Answer 48

@phi he has been given the definition of of odd and even, so it IS kosher.....

Answer 49

I do hope you are not going to disagree with that......

Answer 50

yes:
she gave us a direct proof to an odd number: any integer n1 n=2k+1 for some integer k

unless you have proven 2(n-2)-1 is odd , how can you use it

Answer 51

Because she gave the definition for even as well (ie 1 less or more)

Answer 52

i have to leave for class in 5 mins.....

Answer 53

you would have to prove that not even means odd. you only have the two definitions, and assuming they are complete without proof is not kosher

Answer 54

Actually, the teacher's intent is quite clear, she signalled the route for the proof...

Answer 55

The definitions are GIVEN, they do not need proof.

Answer 56

Here is how I would do it, sticking with the definition of odd


given 2n-5, n an integer

show that m= 2n-5 is odd

Definition of odd: m= 2k+1 for some integer k

find integer k:
2n-5= 2k+1
2k= 2n-6
k= n-3
n-3 is an integer (both n and 3 are integers, and integers are closed under subtraction)
so k is an integer
also 2k+1 = 2n-6+1= 2n-5 = m
having showed that integer k exists, m must be odd

Answer 57

I can also do another way, so what?

Answer 58

i just have to come up with a formal proof to: if n is an integer, then 2n-5 is an odd integer using the fact that an odd integer is n=2k+1 and and even integer is n=2k

Answer 59

And you have done that (I wouldn't exactly call it "formal", but I am convinced)

Answer 60

how would you make it formal?

Answer 61

You don't need to worry about that at this level.

Also, in number theory (sort of like this question) it is quite customary to be less than formal.

Answer 62

so number theory is next?

Answer 63

second or third year university, usually (but you are doing a lot of it anyway, it is just not called that)

Answer 64

Anyway, just write it up neatly and clearly for your teacher (hope she likes it:-)

Answer 65

i gotta run, i cant thank you enough for your patience.....thanks yoda, ill see you around :-D

Answer 66

Ciao!