# Lecture - Introduction to University Mathematics, Lecture 2

> Source: https://ollybritton.com/notes/uni/prelims/mt22/introduction-to-uni-maths/lectures/lecture-2/ · Updated: 2022-10-09 · Tags: uni, lecture

> Mainly about the binomial theorem and then a very basic introduction to sets.

### Notes
Defining binomial coefficients. For $n, k \in \mathbb{N}$, then the binomail coefficient is $\left(\begin{matrix}n \\ k\end{matrix}\right) = \frac{n!}{(n-k)!\times k!}$. Other conditions like $0 \le k \le n$ and equal to $0$ for $k > n$.

Famous lemma:

$$
\left(\begin{matrix} n \\ k-1 \end{matrix}\right) + 
\left(\begin{matrix} n \\ k \end{matrix}\right) = \left(\begin{matrix} n+1 \\ k \end{matrix}\right)
$$

Can be proven just by plugging in the definitions and messing around with the algebra.

Can prove the binomial theorem $(x + y)^n = \sum^n_{k=0} {}^n C_k x^k y^{n-k}$ by inducting on $n$. There’s two key steps to the proof:

- Taking out an $x^{n+1}$ from a sum by reducing the limit of the sum from $n$ to $n-1$, and taking out a $y^{n+1}$ by starting from $k = 1$ rather than $k = 0$.
- Making the sums match up by letting $\hat{k} = k - 1$.

Sets are a collection of objects (though this definition isn’t precise enough because it runs into problems such as Russel’s Paradox).

A set $A$ is a subset of $S$ if every element of $A$ is in $S$. This means things like all sets are subsets of themselves. Proper subsets are where they are not equal.

The power set of a set is the set of all subsets of the set.

You can get ordered pairs (and more generally ordered n-tuples). They seem like vectors at first glance but the elements can be anything, e.g. matrices.

$A \times B$ is the cartesian product of two sets, which means the set of all ordered pairs coming from each set.

$A^3 = A \times A \times A$ and consists of ordered 3-tuples.

Russel’s Paradox: Let $H = \{ \text{set } S : S \notin S \}$, e.g. the set of all sets not containing themselves. The question is, is H in itself? If it is, then it shouldn’t be by definition. If it isn’t, then it should be by definition. Like the [Grelling-Nelson Paradox](https://ollybritton.com/notes/random/grelling-nelson-paradox/), and probably came first.

### Flashcards
What does this equal? $$\left(\begin{matrix} n \\\\ k-1 \end{matrix}\right) + 
\left(\begin{matrix} n \\\\ k \end{matrix}\right)$$::
$$
\left(\begin{matrix} n+1 \\\\ k \end{matrix}\right)
$$

What are the two unintuitive steps in proving the binomial theorem, about $(x + y)^n$?::
* Taking $x$ and $y$ terms out of the sums by reducing the limit, and
* Making two seperate sums match up by making an arbitrary change of variable.

What is the cartesian product $A \times B$ of two sets?::
The set of all ordered pairs of elements in $A$ and $B$.

What is the problematic set in Russel’s paradox?::
$$H = \{ \text{set } S : S \notin S \}$$

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