## 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

## Incomplete proof.

f and g are functions of real variables, strictly increasing and strictly decreasing respectively on R (both surjections), I’m asked to prove that there exists at most one solution to the equation f(x)=g(x). My attempt: Suppose f(x)=g(x)=m, for some x∈R⟹∃a,b∈R|a<x<b. Then : f(a)<f(x)<f(b) and −g(a)<−g(x)<−g(b). Hence f(a)−g(a)<f(x)−g(x)<f(b)−g(b). So if we can prove that f(a)−g(a)<0 and … Read more