Solving Discrete Math Questions - Does Integer Set Include 0?

In summary, the conversation discusses the inclusion of 0 in the set of integers and its use in proofs. It also touches on the definition of natural numbers and the use of counter-examples in disproving statements about all integers. The final part of the conversation presents a problem about the sum of three consecutive integers and a proof that it is not always even.
  • #1
EvLer
458
0
I am in discrete math class right now and trying to get the sets of numbers straight.
So, does the set of integers include 0? Is it ok to use 0 in proofs, that makes finding a counter-example a lot easier and disprove a statement about all integers.
Was just wondering if that is legal... feedback on this is very much needed :smile:
 
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  • #2
Yes, 0 is an integer. 0 may or may not be included in your courses definition of the natural numbers though.
 
  • #3
EvLer said:
I am in discrete math class right now and trying to get the sets of numbers straight.
So, does the set of integers include 0? Is it ok to use 0 in proofs, that makes finding a counter-example a lot easier and disprove a statement about all integers.
Was just wondering if that is legal... feedback on this is very much needed :smile:

Are you sure you meant "integers" and not "natural numbers"?

0 certainly is a member of the "integers". When Peano constructed his axioms for the "natural numbers" he included 0 but today, the "natural numbers" is considered equivalent to "positive integers" which does not include 0.

Could you give an example of the "statement about all integers"? If, for example, it say "for all positive integers", then a counter-example involving 0 would not be valid.
 
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  • #4
the statement goes: prove for every integer n...
which i guess they do not mean natural numbers only, so I used 0 as a counter example. Yeah, I am careful with the "positive integers" which would exclude zero (thanks to HallsofIvy).

Thanks all for help everyone.
 
  • #5
HallsofIvy said:
When Peano constructed his axioms for the "natural numbers" he included 0 but today, the "natural numbers" is considered equivalent to "positive integers" which does not include 0.

There's no universal definition of natural numbers today. Some authors include 0, some don't.

EvLer said:
the statement goes: prove for every integer n...

What is the rest of the statement? Is it true for the rest of the integers? If 0 is the only counter example, it may be an oversight (or maybe that's what they want you to be looking out for given your induction question).
 
  • #6
EvLer said:
the statement goes: prove for every integer n...
which i guess they do not mean natural numbers only, so I used 0 as a counter example. Yeah, I am careful with the "positive integers" which would exclude zero (thanks to HallsofIvy).

Thanks all for help everyone.
No, the problem does NOT go "prove for every integer n...
What was in place of ...? That's the crucial part! What were you asked to prove?
 
  • #7
ok, the full problem is worded a bit different from what I said originally:
prove or disprove that sum of any 3 consecutive integers is even.

I disproved it like this in short:
given n,n+1,n+2 => 3(n + 1) and taking n = 0, sum is odd.
 
  • #8
That looks fine by me then.
 
  • #9
It looks like that (dis)proof would work for any even numbers including 0. (2,3,4) would be odd, as would (4,5,6) and so on...

Just to pick nits.
 
  • #10
I would look at it this way : Given n, n+1, n+2, the sum (that I will note S) is 3(n+1).
We all agree on that. Now, if n is even (i.e. n = 2k)

S = 3(2k+1) = 6k + 3 = 0(mod2) + 1(mod 2) = 1(mod2)

if n is odd (i.e n = 2k+1)

S = 3(2k +2) = 6(k+1) will be odd or even depending on the choice of k

Hence , the sum of three any consecutive integers is not always even
 

1. What is a discrete math question?

Discrete math is a branch of mathematics that deals with discrete objects rather than continuous ones. It involves studying mathematical structures and objects that can only take on distinct, separated values.

2. What is an integer set?

An integer set is a collection of whole numbers, including positive, negative, and zero. It can be represented by the notation {…, -2, -1, 0, 1, 2, …}.

3. What does it mean for an integer set to include 0?

When an integer set includes 0, it means that 0 is one of the elements in the set. In other words, 0 is a member of the set and is included in its representation.

4. How do you determine if an integer set includes 0?

To determine if an integer set includes 0, you can simply look at its representation or list of elements. If 0 is listed as one of the numbers in the set, then it includes 0. Another way is to check if 0 is a solution to the equation or problem being presented.

5. Why is it important to know if an integer set includes 0?

Knowing if an integer set includes 0 is important because it can affect the outcome of a mathematical problem or equation. Also, some mathematical concepts and properties, such as the additive identity property, involve the number 0, so it is essential to know if it is included in the set being studied.

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