Recent content by MI5

Fantastic explanation. Thank you.

WOW! All I can say is thanks. (Giggle)

Let $ \left\{A_1, A_2, \cdots , A_n\right\}$ be a system of subsets of a finite set $A$ such that these subsets are pairwise disjoint and their union $A = \cup_{i=1}^{n}A_{i}$. Then $ |A| = \sum_{i=1}^{n}|A_i|$. (1) Proof: According to the hypothesis, each $a \in A$ belongs to exactly one of...

I still don't understand I'm afraid. Could you say bit more please?

Theorem: If c|ab and (b, c) = 1 then c|a. Proof: Consider (ab, ac) = a(b, c) = a. We have c|ab and clearly c|ac so c|a. It's not so clear to me why c|ac. Perhaps I'm missing something really obvious.

Thank you. One more question. This is from the beginning of a proof of the division algorithm (as you probably recognised). The thing is, considering this particular set came out of a thin air. Do you know - or would you be able to give any motivation?

I think I understand now, thank you. Should it be $x \le -500,000$ though?

My question concerns proving the set of non-negative integers of the form $a-dx ~~(a, d, x \in \mathbb{Z}, d \ge 1)$ is nonempty. This is the proof from my book. If $a \ge 0$, then $a = a-d\cdot 0 \in S$. If $a < 0$, let $x = -y$ where $y$ is a positive integer. Since $d$ is positive, we have...