# Tagged Questions

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

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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.)? ...
845 views

### How to prove that $z\gcd(a,b)=\gcd(za,zb)$

I need to prove that $z\gcd(a,b)=\gcd(za,zb)$. I tried a lot, for example, looking at set of common divisors of the two sides, but I can't conclude anything from that. Can you please give me ...
11k views

### 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?
9k views

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 ...
3k views

### 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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### 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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### Proof the inequality $n! \geq 2^n$ by induction

I'm having difficulity solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this with induction. I started like this: The lowest natural number where the ...
3k views

### 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?
21k views

### 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 ...
7k views

### 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 ...
2k views

### Prove that $(a-b) \mid (a^n-b^n)$

I'm trying to prove by induction that for all $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$, that $(a-b) \mid (a^n-b^n)$. The base case was trivial, so I started by assuming that $(a-b) \mid (a^n-b^n)$. ...
3k views

### Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$

Prove that $\int_a^b f(x)\,dx \gt 0$ if $f \geq 0$ for all $x \in [a,b]$ and $f$ is continuous at $x_0 \in [a,b]$ and $f(x_0) \gt 0$ EDIT. Please ignore below. It is very confusing actually -.- ...
### $A \oplus B = A \oplus C$ imply $B = C$?
I don't quite yet understand how $\oplus$ (xor) works yet. I know that fundamentally in terms of truth tables it means only 1 value(p or q) can be true, but not both. But when it comes to solving ...