Introductory explanation of the Big-Oh properties
Let's start by considering the positive numbers. There is a notion of order:
$$ x \leq y. $$
We can add and multiply numbers, and the order relation respect these operations:
$$
x \leq y, z \leq w \Longrightarrow x + z \leq y + w, xz \leq yw.
$$
Big O notation is very similar. We consider, for simplicity, functions which accept a natural number as input, and output a positive real. We define an order relation:
$$
f = O(g) \text{ iff there exists } C &gt; 0 \text{ such that } f(n) \leq Cg(n) \text{ for all } n.
$$
(The funny notation $f = O(g)$ is just a notation for an order relation. Sometimes $f \ll g$ is used instead, but this is not common in computer science.)
This is similar to what we had above, with two important differences:

Since we have a function rather than a single value, we need the inequality to hold for all inputs.

We put in the extra constant $C$. This has the effect of identifying functions which differ only by a constant factor.
In more details, if $Af(n) \leq g(n) \leq Bf(n)$ for all $n$ then $g$ is &quot;equivalent&quot; to $f$ in the sense that $f = O(h)$ iff $g = O(h)$ and $h = O(f)$ iff $h = O(g)$.


The big O order relation also respects addition and multiplication:
$$
f = O(g), h = O(k) \Longrightarrow f + h = O(g + k), fh = O(gk).
$$

For positive numbers, we have not only $\leq$ but also $\geq$. These are related by
$$ y \geq x \text{ iff } x \leq y. $$
The relation corresponding to $\geq$ for functions is $g = \Omega(f)$, defined by
$$
y = \Omega(x) \text{ iff } x = O(y).
$$
This order relation also respects addition and multiplication.

What happens if $f = O(g)$ and $f = \Omega(g)$? When this sort of situation happens in the universe of numbers, that is when $x \leq y$ and $x \geq y$, then the two numbers are equal: $x = y$. In the case of functions, we don't get equality, but rather the notion of equivalence that was mentioned above: $Af(n) \leq g(n) \leq Bf(n)$ for some positive $A,B$. In this case we write $f = \Theta(g)$.
You can think of $f = \Theta(g)$ as an analog of equality. Indeed, it is possible to quotient by this equivalence relation and work with functions &quot;up to constant factors&quot;, but there is no need to understand this point of view when working with asymptotic notation. It suffices to understand that if $f = \Theta(g)$ then, for the most part, $f$ and $g$ are interchangeable.
(They are not interchangeable when appearing in the exponent: for example, $n^2$ and $n^3$ don't have the same asymptotic rate of growth, although $2 = \Theta(3)$. They are also not interchangeable if appearing at the base of a non-constant exponent: $2^n$ and $3^n$ have different rates of growth.)
An important special case is $Cf = \Theta(f)$ (for $C &gt; 0$). That is, in most cases $f$ and $Cf$ are interchangeable. In other words, asymptotic notation hides constant factors. This is one of the advantages of big O notation.

Big O notation satisfies the following peculiar property:
$$
f + g = \Theta(\max(f,g)).
$$
This useful property allows us to simplify expressions: for example, $n^2 + n = \Theta(n^2)$. Generalizing this, if $f$ is a degree $d$ polynomial, then $f = \Theta(n^d)$.