Proofs and Counterexamples

Hello! First post here, so kind of a newbie.

Anyways, I need to state whether these statements are inconsistant (false) or provable (true) and to provide a justification or counterexample:

1) All numbers are even

2) For all numbers x > 0,1/x < x

a. Dogs that don’t ever bark always bite.

b. There is a dog with soft fur and no teeth.

c. Dogs that bite are never friendly and they always have teeth.

d. Dogs that have soft fur are friendly and they never bark.

I would assume some solutions, but I don't think it would provide proper justification...

So, yeah. Thanks in advance! :D

Re: Proofs and Counterexamples

Hi there, not sure if these questions fit under 'Geometry'

What course are you studying? This could influence the way you answer these questions.

In a general sense,

Quote:

Originally Posted by

**Crayola**

1) All numbers are even

To prove this one incorrect you jusy need to find a number that is odd.

Quote:

Originally Posted by

**Crayola**

2) For all numbers x > 0,1/x < x

You could graph both $\displaystyle \frac{1}{x}$ and $\displaystyle x$ and find if this is true.

Or maybe just pick some numbers to test for x? Is the inequality true for very large x?

Re: Proofs and Counterexamples

Quote:

Originally Posted by

**Crayola** Hello! First post here, so kind of a newbie.

Anyways, I need to state whether these statements are inconsistant (false) or provable (true) and to provide a justification or counterexample:

**1) All numbers are even**
2) For all numbers x > 0,1/x < x

a. Dogs that don’t ever bark always bite.

b. There is a dog with soft fur and no teeth.

c. Dogs that bite are never friendly and they always have teeth.

d. Dogs that have soft fur are friendly and they never bark.

I would assume some solutions, but I don't think it would provide proper justification...

So, yeah. Thanks in advance! :D

Suppose that all numbers are even, so the are only one prime, and this is false! (Giggle)

Re: Proofs and Counterexamples

I think he has to say if the set of assumptions are consistant or not.

So is {1), 2)} consistant ? Is {a., b., c., d.} ?

For 1), 2) : i assume that by 'numbers', you mean $\displaystyle \mathbb N$.

As 1) $\displaystyle \Rightarrow$ 2) (let x>0, and by 1), x is even so $\displaystyle x = 2p$ with p > 0, so $\displaystyle x^2 = 4p^2 > 1$), we may be tempted to say that it's consistant (because it's resuming to 1) only).

But the Peano axioms (defining $\displaystyle \mathbb N$) can not coexist with "1 is even" (which is said by 1)).

So it is not consistant for me. Maybe you can precise a little more the context.

For a., b., c., d. : this is a easy one.

b. say there is one dog with soft fur and no teeth.

So, by d., this dog is friendly and never barks.

So, by a., he always bites.

So, by c., he is not friendly and has teeth : contradiction with the fact that he is friendly (line 2) and has no teeth (line 1).