Finding counterexamples

[benderzfloozy]

Hi. There are two statement in which I am to find a counterexample. The first:

The number n is an odd integer if and only if 3n+5 is an even integer.

Here, I assumed that n is even such that n = 2k. I inserted it into 3n+5 and came up with 6n+5. There's nothing factorable, so 3n+5 is not odd when n is even. I then tried doing the opposite, saying that n is odd. I inserted n = 2k+1 into 3n+5 and got 6n+8. 6n+8 is factorable: 2(3n+4). So, 3n+5 is even when n is odd. I can't seem to find a counterexample!

The second statement: The number n is an even integer if and only if 3n+2 is an even integer. So, I assumed the opposite. I plugged n = 2k+1 into 3n+2, and came up with 6k+5. So, when n is odd, 3n+5 is odd.

I can't seem to find a counterexample for either! Any insight is GREATLY appreciated!

 

[stapel]

I'm afraid I don't follow your reasoning...?

To show "If n is odd, then 3n + 5 is even", let n = 2k + 1 for some integer k. What is the result? Can you factor out a 2?

To show "If 3n + 5 is even, then n is odd", let 3n + 5 = 2k + 6 for some integer k. What then is the parity of 3n? What then must be the parity of n?

Eliz.

 

[benderzfloozy]

What I'm trying to do is find a counterexample to "If n is odd, then 3n + 5 is even" and for "If n is even, then 3n + 2 is even".

What I've done was assume that n was odd, thus getting n = 2k+ 1. I plugged it into 3n + 5 and came up with 6k + 8. A two is factorable, but I didn't think that this disproved the statement.

I'm trying to find out how to disprove both statments, but I just end up proving it. Thanks.

 

[pka]

counterexample. The first:
The number n is an odd integer if and only if 3n+5 is an even integer.
The number n is an even integer if and only if 3n+2 is an even integer. So, I assumed

Has is occurred to you that an odd number times 3 is odd and the sum of two odd numbers is even?

Has is occurred to you that an even number times 3 is even and the sum of two even numbers is even?

 

[stapel]

What I'm trying to do is find a counterexample to "If n is odd, then 3n + 5 is even" and for "If n is even, then 3n + 2 is even".

Why?

Eliz.

 

[benderzfloozy]

That's what the problem is; to find a counterexample and not to prove it. That's what I've been having trouble with.

 

[daon]

That's what the problem is; to find a counterexample and not to prove it. That's what I've been having trouble with.

Counter-examples to a true statement shouldn't exist! Sounds like a "trick question".

 

[stapel]

That's what the problem is; to find a counterexample....

Since these are true (that is, non-falsifiable) statements (which is why proofs can be, and have been, outlined for them), you're gonna be a long time finding counter-examples.

Eliz.

 

[benderzfloozy]

Thanks. I kind of figured, because I assumed that n was both odd and even, and it came out true. I guess I was just getting more confused the longer I tried!