Prove that $X\triangle\emptyset=X$

I’m working on my proofs involving sets, though this one is not a homework problem, so if you wish to provide your own example, so be it. I am working on exercise 3.3.14 (1) in Bloch’s Proofs and Fundamentals. It asks me to prove that $X\triangle \emptyset=X$. The following is what I have so far … 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