Need help with two simple proofs

[eku_girl83]

Here's my problem:
Provide either a proof or a counterexample for each of these statements.
a) For all real numbers x and y, if x is greater than 1 and y is greater than zero, then y^x is greater than x.

My proof:
Suppose x is some real number greater than 1 and y is some real number greater than 0.
Suppose x=2 and y=1/4.
Then y^x=(1/4)^2=1/16 and 1/16=y^x is less than x=2.
Now suppose that x=3 and y=2.
Then y^x=2^3=8 and 8=y^x is greater than x=3.
Hence if x is greater than 1 and 0 less than y less than or equal to 1, then y^x is less than or equal to x.
But if x is greater than 1 and y is greater than 1, then y^x is greater than x.
Therefore the statement "if x is greater than 1 and y is greater than 0, then y^x is greater than x" is not true for all real numbers x and y.

Is this a good proof? How can I improve it or make it clearer?

b) For integers a, b, c, if a divides bc, then either a divides b or a divides c.

I'm not really sure where to go with this one, so hints would be welcome.
I do know that if a divides bc, then bc=ak, where k is a natural number.
Similarly, a divides b means that b=aj and a divides c means that c=ai, where j and i are also natural numbers.
Which proof techinique do I use here? contradiction, contraposition, or direct proof?

Thanks ahead of time,
eku_girl83

 

[TenaliRaman]

your first proof looks solid ..
what u have done is simply give a counter example
u could have chosen some simpler values say,
x=2 and y=0.1 so (0.1)^2 = 0.01 < 2
QED

your second is again a counterexample one,
a = 12 b = 4 and c=6
QED

-- AI

 

[matt grime]

1) A proof is something that must be true for all x and y satisfying those constraints.
So ypu've not proven the statement. You have found a counter example. I do'nt understand why after finding a counter example you do something else too.
Your deduction

"But if x is greater than 1 and y is greater than 1, then y^x is greater than x."

based upon those two examples is also not true.

2) do you think it's true? hint that is sometimes used as a definition for what it means fo a to be a prime.

 

[TenaliRaman]

if u ask me
a proof is either the one that validates the statement or invalidates it

so i still accept that as a proof.

Usuall such questions are tagged with,
" prove or disprove blah blah blah..."

-- AI

 

[matt grime]

It asked for a proof that the statement is true or a counter example.
thus we may take the posters use of the word "proof" to indicate proving it true, when they post a counter example instead. (Ie prove it false), but the usage of the word is poor, and confusing, especially given that after giving a counter example, they then "prove" using one example a false statement.