# Topology: How to formally write proofs?

• Dec 3rd 2012, 05:53 PM
Topology: How to formally write proofs?
Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation?

A simple example: Prove that $\displaystyle A\in 2^A$

Another simple example: Prove that $\displaystyle A \subset B \iff A \cup\B = B$

• Dec 5th 2012, 05:45 AM
ModusPonens
Re: Topology: How to formally write proofs?
You have to use english, or else you would have to resort to symbolic derivations in some logic. See the examples of proofs in a topology book and you'll get the style.
• Dec 5th 2012, 06:39 AM
HallsofIvy
Re: Topology: How to formally write proofs?
Quote:

Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation?

A simple example: Prove that $\displaystyle A\in 2^A$

The standard way to prove that $\displaystyle x\in U$ is to show that x meets the conditions given in the definition of U. What is the definition of $\displaystyle 2^A$?

Quote:

Another simple example: Prove that $\displaystyle A \subset B \iff A \cup B = B$
The standard way to prove "$\displaystyle X= Y$" is to prove both $\displaystyle X\subset of Y$ and $\displaystyle Y\subset of X$. And the standard way to prove "$\displaystyle X\subset Y$" is to start "if $\displaystyle x\in X$" and use the definitions of both X and Y to prove that "$\displaystyle x\in Y$".

For example, to prove "if $\displaystyle A\cup B= B$ then $\displaystyle A\subset B$" start by saying "if $\displaystyle x \in A\cup B$" and argue that, then, $\displaystyle x\in A\cup B$ and so, because $\displaystyle A\cup B= B$, $\displaystyle x\in B$. Since every member of A is a member of B, $\displaystyle A\subset B$. I'll leave the other way (which is a little harder!) to you.

Quote:

(Your B in $\displaystyle A\cup\B$ did not show up because you had "\" in front of the B and Latex does not recognize "\B".)
• Dec 5th 2012, 07:20 AM
Deveno
Re: Topology: How to formally write proofs?

A ⊆ B implies AUB = B.

the way we show two sets are equal is to demonstrate they contain the same elements. so if we start with A ⊆ B, we need to show that everything in AUB is in B, and that everything in B is in AUB.

now everything in B is always in AUB (by the definition of union). so the "hard part" will be showing that everything in AUB is in B.

you'd start like so. suppose x is in AUB = {x in T : x is in A or x is in B} (here, T is "some set" that A and B both belong to, our "universe of discourse").

so either x is in A, or x is in B.

if x is in B, then certainly x is in B.

on the other hand, if x is in A, then.....(you should use something about the relationship between A and B here, now)