4. Well what about Size
● Can you think of things that have an
infinite size?
● Let’s say two collections have the same
size if there exists a
“Bijection” between them. I.e. for each
element in one set you have exactly one
element in the other
5. Counting
Say that a set is infinitely countable if it has the same size as
the natural numbers.
I.e. there exists a strategy for labelling each element of
your collection by a whole number
Examples of countable sets?
6. Counting Pairs
Collections made of pairs of
elements from countable sets
are also Countable
So all rational numbers (fractions)
Are countable
7. So how big are The “REAL”
numbers?
● Well, you can’t count them, no matter how hard you try
(please don’t)
● Another thing that might seem just as weird is that the
Real numbers have the same size as the interval (0, 1)
thanks to our old friend sigmoid
8. Binary Search
● Given a sorted array of size n find the index of a given
element in the array
● Each element in array is uniquely determined by the
steps of the binary search used to find its index
sidenote* good for debugging code, because it’s fast
9. Infinite binary Search!!!!
So at every step of the search lets
give the element a 0 if it ends up
on the left and 1 if ends up on the
right
Where do numbers starting with
(0,1,0,1,....) lie?
10. Cantor’s diagonalization
Argument
Suppose there was a strategy
For counting the unit interval,
Then you could construct a new
binary sequence that has not
Been counted, which leads to a
contradiction