Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in f^{-1}(X)$. Since $X \subset Y$, then $y \in Y$. If $y \in Y$, then $f^{-1}(y) \in … Read more

Proving 10\cdot n=010\cdot n=0 for all n\in\mathbb{Z}n\in\mathbb{Z} with n\geq 0n\geq 0 using strong induction

The question says 10\cdot n=0 for all n\in\mathbb{Z} with n\geq 0. Here is my proof by strong induction: Base case: 10\cdot0=0. Let k\geq 0, and suppose that for any m\leq k we have that 10\cdot m=0. Consider 10\cdot(k+1). The number k+1 can be written as m+l for some numbers 0\leq m,l\leq k. By the induction … Read more

Prove that there exist four distinct real numbers a, b, c, d such that exactly four of the numbers ab,ac,ad,bc,bd,cd are irrational

So i am doing this example and i found a way to prove there exists that but it seems ugly. Can you guys help me find a better solution? My proof: a*b is irrational when one of them is irrational and the other is rational. Assume one of the numbers is irrational (say “a”). We … Read more

What is the most basic way to show that $\emptyset \in S$

Let $S$ be a set, what is the most basic way to show that $\emptyset \in S$? I am asking because sometimes a question in involving a topology $\tau$ or a $\sigma$-algebra $\Sigma$ will want you to show that $\emptyset$ is in $\tau$, $\Sigma$ etc. And most proofs glosses over this. For example, let $\Sigma$ … Read more

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset A⊆M⟹A is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy Def: A is sequentially compact if every sequence has a convergent subsequence. By contradiction, assume A is a sequentially compact subset of M and … Read more

Prove Without Induction: n∑k=21k(k−1)=1−1n\sum\limits_{k=2}^{n} \frac{1}{k(k-1)} = 1 – \frac{1}{n} [duplicate]

This question already has answers here: Showing ∑ni=11i(i+1)=1−1n+1 without induction? (4 answers) Closed 5 years ago. everybody. I’m suppose to prove this without induction: Prove Without Induction: n∑k=21k(k−1)=1−1n I’m not sure how to do it. I tried a bit of algebraic manipulation, but I’m not sure how to do it. It’s suppose to be basic. … Read more

Show that $[0,1)$ has no maximum, i.e. $\not \exists \max[0,1)$

Show that $[0,1)$ has no maximum, i.e. $\not \exists \max[0,1)$ My Attempted Proof Assume $\max[0,1)$ exists and put $\alpha = \max[0,1)$. Now $\alpha < 1$ else $\alpha \not \in [0,1)$. Put $\gamma= 1- \epsilon$ where $0 < \epsilon \leq 1$. Then $\gamma > \alpha$ for small enough $\epsilon$ and $\gamma \in [0,1)$. Reaching a contradiction. … Read more

How to relate areas of circle, square, rectangle and triangle if they have same perimeter??

I was given a question which was like: Suppose that a circle, square, rectangle and triangle have same perimeter. How area their areas related?? My work: I broke the question in parts and tried to prove it seperately: STEP 1. Suppose the given perimeter is P.So the radius of circle will be r=P/2π. And hence … Read more

Proving that the graph of a convex function lies above its asymptote

Suppose that a function f:(a,∞)→R, where a∈R, is twice differentiable and convex and that f has an asymptote at +∞. I want to prove that the graph of f never goes below the asymptote. The problem seems to be trivial but I can’t solve it. Does anyone have any ideas? Answer Suppose the asymptote is … Read more