1. ## Proving mathematical statements

This is something I have great trouble with. I struggle to prove statements, no matter how obvious they are.

For example,

"Show that the sum of a rational number and an irrational number is irrational."

Intuitively I know this is true. But, how can I show it?

2. Originally Posted by Glitch
This is something I have great trouble with. I struggle to prove statements, no matter how obvious they are.

For example,

"Show that the sum of a rational number and an irrational number is irrational."

Intuitively I know this is true. But, how can I show it?
If you do a bunch of a proofs/read a bunch of proofs, you get used to some common techniques, such as proof by contradiction, combinatorial proof (where appropriate), how to set up variables, which additional lines to add in a geometry proof, etc.

Here, rewrite the rational number as p/q with p and q both integers, q nonzero. If the irrational number is z then their sum is

x = p/q + z

Suppose x is rational, rewrite as a/b like above

a/b = p/q + z

Manipulate

a/b - p/q = z

(aq - pb)/bq = z

We have just shown that z is rational. This is a contradiction. QED.

3. Originally Posted by Glitch
This is something I have great trouble with. I struggle to prove statements, no matter how obvious they are.

For example,

"Show that the sum of a rational number and an irrational number is irrational."

Intuitively I know this is true. But, how can I show it?
You can show it by providing a proof by contradiction.

Let $\displaystyle q\in\mathbb{R}\backslash\mathbb{Q}$ (i.e. q is an irrational number) and let $\displaystyle p\in\mathbb{Q}$ with $\displaystyle p=\frac{a}{b};\,a,b\in\mathbb{Z}$ with $\displaystyle b\neq 0$.

Now, suppose that $\displaystyle \frac{a}{b}+q$ is rational. Then $\displaystyle \exists\,m,n\in\mathbb{Z}:\frac{a}{b}+q=\frac{m}{n }$ with $\displaystyle n\neq 0$.

Then $\displaystyle q=\frac{m}{n}-\frac{a}{b}=\frac{bm-an}{bn}$. Since $\displaystyle \frac{bm-an}{nb}\in\mathbb{Q}$, it follows that $\displaystyle q\in\mathbb{Q}$, a contradiction. Therefore, $\displaystyle \frac{a}{b}+q$ has to be irrational.

Does this make sense?

4. I find that when people have trouble understanding what they need to proof, they don't understand the underlying logic the statement has. If I reworded your problem, I would say something like "Every sum of a rational and irrational number is irrational." This is known as a Aristotelian form. In this case, we have the form "every $\displaystyle x$ is $\displaystyle y$". For example, Every man is mortal. How does this translate into logic? For every $\displaystyle x$, if $\displaystyle x$ is a man, then $\displaystyle x$ is mortal.

Now, we can apply this method to this problem For every $\displaystyle x$ and $\displaystyle y$, if $\displaystyle x$ is rational and $\displaystyle y$ is irrational, then $\displaystyle x+y$ is irrational. Still doesn't sound easy, and I agree a proof by contradiction is best. So we assume the negation, which is "There exists a $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle x$ is rational, $\displaystyle y$ is irrational, and $\displaystyle x+y$ is rational". Then say what it means for $\displaystyle x$ and $\displaystyle x+y$ to be rational (in other words, write out the definition), and say what it means for $\displaystyle y$ to be irrational (write out its definition). This gives you a beginning, and you proceed the others suggest.

Honestly, there are two ways in my opinion to help with your proof writing: Keep doing more proofs and study logic. But even so, it takes time and lots a practice. It took me Real Analysis I, Intro to Abstract Algebra, and Linear Algebra, plus a course in Formal Logic (find that in the philosophy dept.), before I got really comfortable with just basic proof writing. You'll get there in your own time. Just keep trying.

5. In a certain view, proof presupposes axioms and rules of inference. In ordinary mathematics, the axioms and rules of inference might not be explicitly stated in many situations, but one could ordinarily uncover the implicit axioms and rules of inference.

If you don't even know how to APPROACH proving things, I would very strongly suggest learning symbolic logic, in particular the first order predicate calculus. Virtually all mathematical proofs can be done using only certain axioms along with the rules of inference of the first order predicate calculus. Mastering working in the first order predicate calculus will give you a solid foundation (and techniques for) mathematical proofs. Even if your mathematical proofs are not in pure symbolic form, a study of symbolic logic provides you with techniques to use.

For a text, I highly recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar, which is the best textbook I've seen for learning how to work in the first order predicate calculus.

6. Thank you.