# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
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### Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
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### Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
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### Prove the inequality $n! \geq 2^n$ by induction

I'm having difficulty solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this by induction. I started like this: The lowest natural number where the ...
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Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
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### Proof by contradiction vs Prove the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
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### how to be good at proving?

I'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know how ...
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### Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$. [duplicate]

Here is my proof, I would appreciate it if someone could critique it for me: To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, ...
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### How would one be able to prove mathematically that $1+1 = 2$?

Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
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### Sum of cubes proof [duplicate]

Prove that for any natural number n the following equality holds: $$(1+2+ \ldots + n)^2 = 1^3 + 2^3 + \ldots + n^3$$ I think it has something to do with induction?
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### How to check, whether the formula is a tautology

How do we decide whether the formula in predicate logic is a tautology? Is there some universal way to decide? Let's have an example: ...
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### Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
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### Why is it that when proving trig identities, one must work both sides independently?

Suppose that you have to prove the trig identity: $$\frac{\sin\theta - \sin^3\theta}{\cos^2\theta}=\sin\theta$$ I have always been told that I should manipulate the left and right sides of the ...
### Proof that $1729$ is the smallest taxicab number
For homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number. A taxicab number means that it is the sum of two different cubes and can ...
### Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number I've looked at http://math.stackexchange.com/a/19998 It is known that $2^n-1$ can only be prime if $n$ is prime....