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One basic misconception people often have is that the subset relation implies smaller cardinality (i.e. "size"). E.g., all odd integers are integers, so there
by Gabriel54 3y ago
One basic misconception people often have is that the subset relation implies smaller cardinality (i.e. "size"). E.g., all odd integers are integers, so there are more integers that odd integers. But this is not a very useful notion of size. For example, how would we compare the set of multiples of two with the set of multiples of three? We want a notion of size independent of the underlying "objects" in our sets. This is why we define cardinality in terms of mappings between sets.
- hinkley 3y agoAs software engineers this is easier to explain I think. Create a listA with all integers. Create a listB with all multiples of 3. Is listA.length > listB.length? No, it is not.
- jcparkyn 3y agoThat somewhat falls apart when you start comparing infinite sets that actually _do_ have different cardinalities.
- moritzwarhier 3y agoI think the pseudocode might just be hard to write here const N = [1,2,3 ... ]; const countableSet = N.map(n => getEnumeratedFraction(n)); e.g. here the lambda function maps the natural numbers to all fractions (cantor showed how to easily implement getEnumeratedFraction). For all sets of equal cardinality, a pure function exists that transforms N (or another set of the cardinality you wish) when mapping the set members. And conversely, if such a function exists, the sets are of same size. Since the pseudocode is JS-like, you could also write it using Set. The sets are represented in pseudocode as arrays because enumeration is the point here.
- cubefox 3y agoTo say that some sets have the same size as some of their proper subsets also doesn't seem "useful", so usefulness is not a very strong argument.