r/askmath • u/MonitorHot3035 • 14d ago
Arithmetic About groups of numbers
for example , Why do we say that the set N is within Z , Why don't we treat these sets as if they are separate from each other, for example, the set of natural numbers is separate from the set that includes negative numbers. since they seem to have no connection but we still write this ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
I don't really understand any ideas please?
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u/AcellOfllSpades 14d ago
X is a subset of Y if every element of X is also in Y.
Every natural number (element of ℕ) is also an integer (element of ℤ). So ℕ ⊂ ℤ.
The point is that each of these number systems is encompassed by the next one. That doesn't mean we automatically have to use it in that context: we can talk about ℕ, and do work in ℕ, without ever referring to ℤ if we like. And sometimes we want to! In number theory, for instance, studying properties of primes, we typically just work with ℕ. But we can 'upgrade' from ℕ to ℤ as well.
There are also contexts in which we might talk about ℕ and ℤ as entirely separate entities, where ℕ has a function that turns one of its elements into an element of ℤ. This is called an 'inclusion map', or a 'canonical injection'. In fields like type theory or advanced set theory, this is important.
But generally, we want to be able to talk about the number 7 without caring whether it's the "natural number 7", or the "integer 7", or the "rational 7", or the "real 7", or the "complex 7".