Proof with three conditions

Question

Im trying to prove that;

'if x and y are irrational, but x+y is rational, prove that x-y is irrational'

I have tried a proof by contradiction, but I don't know which parts of the question I should negate so that I get a contradiction. Would I assume that the 'P' statement is 'if x and y are irrational, but x+y is rational' and the 'Q' to be 'x-y is irrational'?

Thanks in advance

Answer 1

The statement that you have is :

If $x$ and $y$ are irrational , but $x+y$ is rational, prove that $x-y$ is irrational.

Always remember : what comes before "prove that" is the premise, and what comes after is the conclusion. For contradiction , you must assume that both the premise and the negation of the conclusion are true, and then come up with a statement which is both true and false.

For example, a typical proof by contradiction would look like:

Suppose that $x,y$ are irrational, $x+y$ is rational (the premise is true) and $x-y$ is rational (the negation of the conclusion is true).

Therefore, since the sum of two rational numbers is rational, $(x+y) + (x-y)$ is a rational number. Therefore, $2x$ is a rational number.

However, twice an irrational number is also an irrational number. Therefore, if $x$ is irrational, so is $2x$. Thus, $2x$ is an irrational number.

The statement "$2x$ is rational" is therefore both true and false, and since this cannot happen, the negation of conclusion cannot be true, therefore the conclusion is true, and hence the statement follows.

Always remember : words such as "then", or phrases like "prove that" are useful in distinguishing premise from conclusion(In well-written theorems). Once you do this separation, then you can use the method of contradiction like I did above.

Also note that to us contradiction, the conclusion should be easy to negate, or at least the negation must be manipulable, even if it is difficult. For example, you can't say that the sum of two irrationals is irrational, while you can do this for rationals. Hence, negation was suitable here.

Also, note that negation may be difficult in certain cases. A case in point is the famed $\epsilon-\delta$ definition of continuity. Every Michael,Madan and Kamarajan has issues inverting the statement of continuity at least initially, before they realize how to do it.