The binary relation is reflexive and transitive but is not a symmetric relation. For example, while . Therefore
There are consequently no equivalence classes for this binary relation in the requested sense.
Treat as fixed positive integers. For , . Raising this inequality to the positive power gives
The factor is independent of , as required by Big O notation. Hence
A bound uniform over varying is not what the notation here asserts.
The exponential function eventually grows faster than the fixed polynomial. A direct justification uses the ratio :
For all sufficiently large , this ratio exceeds , so grows without bound. The proposed Big O notation bound is therefore impossible:
The logarithm laws give
For , both and . The expression is therefore at most . Consequently
The Big O notation concerns eventual behaviour; the zero value of at is irrelevant.
The logarithm identity rewrites the expression as . Since , we have , and for ,
Thus the Big O notation bound holds with :
The proposed inverse function cancels on both sides:
The uniqueness of an inverse function therefore gives
The operation applied last must be undone first.
For the bijections and used above, sends to . Its inverse function sends to , whereas sends to . Hence
The inverse factors are generally in the wrong order. For a counterexample, take the bijections and , with inverse functions and . Then
Thus
For particular bijections that commute the equality can hold; the claim fails as a universal assertion.
The function is injective, also called one-to-one, when equal outputs force equal inputs:
Equivalently, different elements of the domain of a function always have different images.
The function is surjective, also called onto, when every element of its codomain has at least one preimage:
The condition concerns the specified codomain, not just the formula defining the function.
Use the symmetric difference formula from part (b), followed by the set difference identity from part (c)(ii):
Taking both complements swaps the two ways to belong to exactly one set, leaving their union unchanged:
An algebraic way to prove the set identity is to show that is a complement of a set for . By the distributive property,
and
A complement of a set is unique: if both complement , then , and similarly . Consequently
Apply the distributive property to the intersection with :
In the first line the repeated factor changes nothing. Rewriting in terms of set differences yields
Write each set difference as an intersection with a complement of a set. Applying De Morgan's laws and then the distributive property gives
The complements turn the inner union into an intersection and vice versa. Thus

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact