# Thread: Define A, B in this proof

1. ## Define A, B in this proof

I need to figure out what my A is and what my B is.

The question is:

Prove by contrapositive that 1+sqrt(2) is irrational. So how do I define my A and B.

all I can think of is A: consider two numbers 1 and sqrt(2)
B: their sum is irrational.

I can prove it but I feel like my A and B are completely wrong. Thanks so much

2. ## Contrapositive

Hello meg0529
Originally Posted by meg0529
I need to figure out what my A is and what my B is.

The question is:

Prove by contrapositive that 1+sqrt(2) is irrational. So how do I define my A and B.

all I can think of is A: consider two numbers 1 and sqrt(2)
B: their sum is irrational.

I can prove it but I feel like my A and B are completely wrong. Thanks so much
$\displaystyle A$ is "$\displaystyle \sqrt{2}$ is irrational"; $\displaystyle B$ is "$\displaystyle (1+\sqrt{2})$ is irrational".

The contrapositive of $\displaystyle (A \Rightarrow B)$ is $\displaystyle (\neg B \Rightarrow \neg A)$.

$\displaystyle \neg B \Rightarrow \neg A$ is If "$\displaystyle (1 + \sqrt{2})$ is not irrational, then $\displaystyle \sqrt{2}$ is not irrational". This is correct, since $\displaystyle (1+x)$ is rational $\displaystyle \Rightarrow x$ is rational.

Hence $\displaystyle A \Rightarrow B$, and $\displaystyle (1+\sqrt{2})$ is therefore irrational, since $\displaystyle A$ is true.