# Tag Info

### Is this proof about independence of events rigorous?

This is not a proof but rather an intuitive explanation of what is happening behind the curtains. While the reasoning is sound, it does not prove the statement as it lacks the unambiguous formalism to ...
Accepted

### Exercise 3.6.8 from "How to prove it"

For your $A \subseteq U$, define $B = U \setminus A$. Then for any $C \subseteq A$: $$C \cap B = C \cap (U \setminus A) = (C \cap U) \setminus (C \cap A) = C \setminus (C \cap A) = C \setminus A$$ ...

### Exercise 3.6.8 from "How to prove it"

You're unlikely to be able to complete the proof if you have no idea what $B$ should be. At this point (or even before you start trying to write the proof), consider some examples, try to visualize ...
Accepted

### Can we use Prove by Contradiction to prove a statement FALSE?

That works, because if you have a statement $P$ and you want to show $P$ is false, you can do proof by contradiction with the statement $Q=\neg P$. You deduce $Q$ is true, so you deduce $P$ is false. (...

### Prove that $x^3 = x^2$ has exactly two solutions

On one hand we know that $1^3=1^2$ and $0^3=0^2$ so $0$ and $1$ are solutions. On the other hand $$x^3=x^2\implies x^3-x^2=0\implies x^2(x-1)=0$$ Therefore, if $x$ is a solution then $x$ must be ...
Accepted

### Prove that $x^3 = x^2$ has exactly two solutions

It obviously proves that $z=1$ is possible, but how does it prove that $z=0$ is possible? No, that is not what the proof you cite is proving. The proof is not saying that $z=1$ is possible, nor is it ...

### Suppose $A$ and $B$ are two sets with $B\subset A.$ Let $f:A\to B$ be injective. Then show that $\exists$ a bijection $h:A\to B.$

The above alternative proof is justified/valid only under the assumption, that we are working with finite sets. To elaborate, the proof might be true in general, but it's not a complete one. It is ...
Accepted

### How to quote definitions concisely?

The negative of any even integer is even. Correctly applying a definition does not necessarily involve quoting the thing. For example, here's a sample proof: Let $n$ be an arbitrary even integer. ...
Accepted

### How to know if a variable is an integer when writing proof?

An integer multiplied by an integer, is an integer. This is because the set of integers is closed under multiplication. In your case, we have $2r^2$, because $r$ is an integer we get that $rr=r^2$ is ...

1 vote

### Is a construction/definition a premise in a logical argument that a proof is?

In general, you can think of the "premises" of the proof to be solid and unchangeable facts that can be used as necessary. However, more of these "premises" can be constructed as ...
1 vote

### To prove that for every $x$, $(x\in Z\implies x\in R),$ is it ok to write "For any $x,$ suppose $x\in Z$. Then... Then $x\in R$"?

Your proof is fine. For a slightly more precise wording I would recommend starting with Let $(x,y)\in ...$ and then proceeding exactly as you have. NOTE Despite some of the comments about "let&...
1 vote

### Disproof: there exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime.

Prove the following claim is true: "$P$" Prove by contradiction, assume "$\neg P$" is true. Prove the following claim is false: "$Q$" Prove by contradiction, assume "...
1 vote
Accepted

### Need a proof for a lemma (Stewart. Galois Theory. Lemma 5.14, about the dimension of simple extensions)

I'm not sure either what they meant by a restatement of lemma $5.9$. Let's look at lemma $5.14$. Consider the minimal polynomial of $\alpha$ over $K$, denoted by \$f^{\alpha}_K := f = \sum^n_{i=0} b_i ...

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