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# the negative of the proposition every natural number is an integer

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### Mohammed

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## Exercise 4.4.3

Exercise statement Fill in the gaps marked (why?) in the proof of Proposition 4.4.4. Proposition 4.4.4. There does not exist any rational number \$latex x\$ for which \$latex x^2 = 2\$. Hints None. How to think about the exercise When we want to show that a natural number cannot be both even and odd, here's…

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## Exercise statement

Fill in the gaps marked (why?) in the proof of Proposition 4.4.4.

Proposition 4.4.4. There does not exist any rational number for which .

None.

## How to think about the exercise

When we want to show that a natural number cannot be both even and odd, here’s a mistake you might make: you might say that if is both even and odd, then there is a natural number such that and . So , which means , a contradiction. This is wrong! You cannot assume that the that acts as a “witness” for the evenness property is the same as the that acts as a “witness” for the oddness property. The most we can say is that for some natural numbers and (without any assumption that ).

Once you have , you might say so that is a natural number. Then you might prove by induction that for every natural number . If you remember back to Chapter 2, this is precisely the example given in the book after Remark 2.1.10. In fact, now that we have defined notions such as “integer”, “half-integer”, and “0.5”, this is actually a valid proof. However in my opinion, this is kind of an “impure” proof. The evenness/oddness properties are defined for natural numbers, and the definitions of these properties mention only natural numbers. This means that any fact we can prove about these properties can sort of be done as if we never knew what integers or rational numbers are. The proof I give below does use subtraction (which we never defined for natural numbers), but if you look closely, it uses only “virtual subtraction” (subtraction where we never get negative numbers) so I consider it a more “pure” proof.

## Model solution

Every natural number is either even or odd, but not both: We first show using induction that every natural number is even or odd. For the base case, we see that so is even. Now suppose inductively that is even or odd. We have to show that is even or odd. If is even, then for some natural number . Thus so is odd. On the other hand, if is odd, then for some natural number . Thus so is even. In either case, we have shown that is even or odd, so this closes the induction.

Now we show that a natural number cannot be both even and odd. Suppose for sake of contradiction that there is some number which is both even and odd. Thus we have for some natural number and for some natural number . So . Now we have two cases, or . Suppose first that . Then implies . Since , we see that is a natural number. Thus by Corollary 2.2.9 we have , a contradiction. Now suppose that . Then so . Since is positive, so we can add to both sides to get . This means we have , so , a contradiction. In either case we arrive at a contradiction, which means our assumption that is both even and odd was false.

If is odd, then is also odd: If is odd, we can write for some natural number . This means . Thus for we have , which shows that is odd.Since , we have : Suppose for sake of contradiction that . Then since are positive, we can multiply on both sides by to get and multiply by on both sides to get , so that . Also since is positive, so is , so , which means that we can add to both sides to get . Putting these two inequalities together, we have , which implies that , a contradiction.

### RELATED

Exercise 4.4.1 May 2, 2020 In "Solutions" Exercise 2.2.3 March 30, 2020 In "Solutions" Exercise 5.5.3 May 15, 2020 In "Solutions"

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Write negation of:

Question

All  natural numbers are integers  and all integers are not natural numbers.

A

B

C

D

## All natural numbers are not integers and some integers are natural numbers.

Medium Open in App Solution Verified by Toppr

Correct option is D)

the negation of

All  natural numbers are integers  and all integers are not natural numbers. is All natural numbers are not integers and some integers are natural numbers. or Some natural numbers are not integers and some integers are natural numbers

more or less option d is correct.

D

3 1

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## Negation of the statement “Every natural number is an integer”.A. All

Negation of the statement “Every natural number is an integer”.A. All-natural numbers are whole numbers.B. Every natural number is not an integer.C. Every natural number is not a real number.D. None of the above.. Ans: Hint:- We h...

Negation of the statement “Every natural number is an integer”.

A. All-natural numbers are whole numbers.

B. Every natural number is not an integer.

C. Every natural number is not a real number.

D. None of the above.

Last updated date: 18th Jan 2023

• Total views: 278.7k • Views today: 7.76k Answer Verified 278.7k+ views

Hint:- We had to only place not such that the resultant statement has the opposite meaning of the given statement. Like the negation of the statement “Every triangle has three sides” can be “Not every triangle has three sides”.

Complete step-by-step solution -

As we know that the negation of any statement is the opposite of the given statement in terms of meaning.

So, to find the negation of the statement we had to place not in the statement such that it makes the given statement false.

Now there can be more than one negation of the given statement because there are many ways to grammatically express any statement.

And the negation of a statement can be logically incorrect also.

So, now let us find the negation of the statement “Every natural number is an integer”

Its negation can be,

“Every natural number is not an integer.”

Or

“It is false that every natural number is an integer.”

Or

“It is false to say that every natural number is an integer.”

Or

“It is not the case that every natural number is an integer.”

Hint:- Whenever we come up with this type of problem where we are asked to find the negation of the given statement then the trick behind the negation statement is, we had to use any of the keywords like false, not, can’t be true, not the case etc. Such that the meaning of the given statement changes or we can say that the given statement becomes false. And we should remember that there can be more than one negation possible for any statement but the meaning of all the negations will be the same because one statement can be written in many ways in grammar.

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Mohammed 9 day ago

Guys, does anyone know the answer?