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.
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
Another simple example: Prove that
thanks for your time.
The standard way to prove that is to show that x meets the conditions given in the definition of U. What is the definition of ?
The standard way to prove " " is to prove both and . And the standard way to prove " " is to start "if " and use the definitions of both X and Y to prove that " ".Another simple example: Prove that
For example, to prove "if then " start by saying "if " and argue that, then, and so, because , . Since every member of A is a member of B, . I'll leave the other way (which is a little harder!) to you.
(Your B in did not show up because you had "\" in front of the B and Latex does not recognize "\B".)thanks for your time.
something to help you proving:
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)