Lecture - Introduction to University Mathematics, II
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]]N, and probably came first.
Flashcards
What does this equal?
\[\left(\begin{matrix} n \\\\ k-1 \end{matrix}\right) +
\left(\begin{matrix} n \\\\ k \end{matrix}\right)\]
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 $B$.