MIT has a portal where they put lots of courses on MANY subjects, and in a less informal way than Kan. Their material is really good, and you will probably find those topics in some of the courses there.
http://ocw.mit.edu/index.htm
EDIT: As mentioned in the comment by eccstartup coursera is another option, but with only a fraction of what you can find on MIT's opencourseware, but it still might have many different courses.
The explanation goes back to the origins of Boolean algebra. Define a "Boolean" variable $\, x \,$ as one where $\, x^2 = x. \,$ For these variables $\, 1-x \,$ denotes logical negation and $\, xy \,$ denotes logical and. Finally, by De Morgan's laws,
$\ 1-(1-x)(1-y) \,$ denotes logical or.
In the context of subsets of a univeral set $\, U, \,$ any subset $\, A \,$ of $\, U \,$ is identified by its indicator function $\, A(x) \,$ which is Boolean valued. Now the number of elements of $\, A \,$ is $\, |A| = \sum A(x) .\,$ Next, the indicator function of $\, A \cap B \,$ is $\, A(x)B(x), \,$ of complement of $\, A \,$ is $\, 1-A(x), \,$ and of $\, A \cup B \,$ is $\, 1-(1-A(x))(1-B(x)). \,$
Given any subsets $\, A_1, A_2, \dots, A_n \,$ and their indicator functions $\, A_1(x), A_2(x), \dots, A_n(x), \,$ we can write
$\, 1-(1-A_1(x))(1-A_2(x)) = A_1(x) + A_2(x) - A_1(x)A_2(x) \,$ which implies $\, |A_1 \cap A_2| = |A_1| + |A_2| - |A_1 \cup A_2|. \,$ This obviously generalizes to any finite number of subsets and is
one form of the PIE.
Now suppose that $\,A = A_1=A_2=\dots=A_n.\,$
Use the binomial theorem to get
$$ (1-A(x))^n = \sum_{k=0}^n {n\choose{k}} (-1)^k A(x)^{k}$$
but since $\,A(x)\,$ and $\,1-A(x)\,$ are Boolean valued
and if also $\,n>0\,$ this simplifies to
$$ 1-A(x) = (1-A(x))^n = 1 + A(x)\sum_{k=1}^n {n\choose{k}} (-1)^k.$$
If now $\,A(x)\ne 0\,$ then this implies that
$$ -1 = \sum_{k=1}^n {n\choose{k}} (-1)^k \qquad \text{ and } \qquad
0 = \sum_{k=0}^n {n\choose{k}} (-1)^k . $$
Best Answer
You might also consider presenting one of the topics in your list in a very different way. I have in mind the treatment of the sieve method (inclusion/exclusion) by Herbert S. Wilf in section $4.2$ of his book generatingfunctionology; a PDF of the second edition is freely available at his site. It’s very pretty and very different from the usual treatment.