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Proof by contradiction

Hi guys, I need some help with this question:
"Use proof by contradiction to show that, given a rational number a and an irrational number b, a-b is irrational"
Any ideas on how to do this

Reply 1

RDKGames

Study Forum Helper

Original post by DankoMan

Hi guys, I need some help with this question:
"Use proof by contradiction to show that, given a rational number a and an irrational number b, a-b is irrational"
Any ideas on how to do this

We have that

a = \dfrac{p}{q}

with

p \in \mathbb{Z}

and

q \in \mathbb{N}

since it's rational. Now we also have that

b

cannot be expressed in that way.

Suppose, on the contrary, that

a-b

is rational.

Then we have that

a-b = \dfrac{r}{s}

with

r \in \mathbb{Z}

and

s \in \mathbb{N}

, then what is

b

(edited 5 years ago)

Reply 2

3pointonefour

Original post by RDKGames

We have that

a = \dfrac{p}{q}

with

p \in \mathbb{N}

and

q \in \mathbb{Z}

since it's rational. Now we also have that

b

cannot be expressed in that way.

Suppose, on the contrary, that

a-b

is rational.

Then we have that

a-b = \dfrac{r}{s}

with

r \in \mathbb{N}

and

s \in \mathbb{Z}

, then is

b

Damn that set notation though

Reply 3

RDKGames

Study Forum Helper

Original post by 3pointonefour

Damn that set notation though

Doesn't help that I initially put them in the wrong sets without realising which allowed the denominators to be zero. :facepalm2:

Reply 4

MR1999

Original post by DankoMan

Hi guys, I need some help with this question:
"Use proof by contradiction to show that, given a rational number a and an irrational number b, a-b is irrational"
Any ideas on how to do this

RDKgames' comment is probably what you want but this is more a general way to do these. The first thing you need to do is work out which statement you're working with in the proof. It helps if you write the original statement in IF-THEN form. That is, If ..., then...

In this case, we have If a is rational and b is irrational, then a-b is irrational. Then what you have to do is negate the THEN part of the statement. That is the statement becomes If... then not...

In this case, we are working with If a is rational and b is irrational, then a-b is rational.

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Proof by contradiction

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