Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
You are not logged in.
Post a reply
Topic review (newest first)
I think I've seen that before, but, then it seems that N is not a subset of Z.
No, is constructed by means of either Dedekind cuts or equivalence classes of Cauchy sequences of rationals. The process is very different and much more complicated because you are now constructing an uncountable set from a countable one.
Thanks,by the way is R constructed the same way?
It depends on how you are defining your sets of numbers. If you are defining them by axioms, then there is nothing to prove since the axioms will include commutativity and associativity of addition. But if you are constructing them from smaller sets of numbers, then commutativity and associativity need to be proved.
For example, here is briefly how is constructed from . Define a relation on by iff . (Hint: Think of as the "difference" .) Then is an equivalence relation and is defined as the set of all equivalence classes under . Let the equivalence class containing be denoted . Then addition in is defined as
After checking that the operation is well defined, one can proceed to verify that is commutative and associative.
Moreover, zero in is the equivalence class and multiplication in is defined as .
Similarly can be constructed from as equivalence classes of the equivalence relation on defined by iff . Letting the equivalence class containing be denoted addition in is definied as
Never mind i found ways to prove commutative and associative law of addition of natural numbers in internet[though i'd appreciate it if anyone tells me how this is extended to Z,Q,R,..]
I have a younger cousin who wants to know how to prove addition is commutative and associative for whole numbers .but i don't know how to do that,can anyone help?plus it'd be good if anyone can tell me how the proof is extended to other number sets.