Arithmetic
We can apply recursion theorem, from the previous section, to define addition and multiplication on [IMAGE: https://cdn.steemitimages.com/DQmX6JzLRTCyUZ9RbmbnQ7NKRvh2BF9b4Z8ojnNiTtWBcmf/image.png].
Suppose we want a function [IMAGE: https://cdn.steemitimages.com/DQmZ37LB9SgDhqTDrERDYFUhMn1JxVU4ryTRYmXWBhEw5V4/image.png] such that [IMAGE: https://cdn.steemitimages.com/DQmSfyrQRCs94xAtgCmkvAm2SApBgMfC9HKrnZ8A3G2xNef/image.png] is the result of adding 5 to n. Then [IMAGE: https://cdn.steemitimages.com/DQmaG7hznF5wuqBTxcnEnPAUZB7pFsFM2P8eu33s2aUeeNZ/image.png] must satisfy the conditions
[IMAGE: https://cdn.steemitimages.com/DQmYp4K5MT75C7PD6RcEWEj44jKtLyD2x2e2i3JqPwWMgBG/image.png]
The recursion theorem assures us that a unique such function exists.
Definition A binary operation on a set A is a function from [IMAGE: https://cdn.steemitimages.com/DQmc7ijW5LzsjXTMW4tVBCJCiNQa2VouppvYJdaUfEoqePq/image.png] into A.
Definition Addition (+) is the binary operation on [IMAGE: https://cdn.steemitimages.com/DQmX6JzLRTCyUZ9RbmbnQ7NKRvh2BF9b4Z8ojnNiTtWBcmf/image.png] such that for any m and n in [IMAGE: https://cdn.steemitimages.com/DQmX6JzLRTCyUZ9RbmbnQ7NKRvh2BF9b4Z8ojnNiTtWBcmf/image.png].
[IMAGE: https://cdn.steemitimages.com/DQmaCtqUfY63rjYs6dLC5ewK6KKeaLN4SzfnLdqqLFTpzkG/image.png]
Thus when written as a relation
[IMAGE: https://cdn.steemitimages.com/DQmUrLkXzXvUzDLCEyFPr53xH65BrSNLizpBR2PSsAbnAJ6/image.png]
In conformity to everyday notation, we write m+n instead of [IMAGE: https://cdn.steemitimages.com/DQmbRyY8gaiqsn1cg11zbQsvq1tqP8mTiSPVE9MuFSvjkeP/image.png].
Theorem 4I For natural numbers m and n,
[IMAGE: https://cdn.steemitimages.com/DQmVcM7ciUqmABgHZ3EjUFmCYdQLdvGFYiTuZNe4NRqigy5/image.png]
This theorem is an immediate consequence of the construction of [IMAGE: https://cdn.steemitimages.com/DQmYWTscg1Sje4MaezGaQ84MhTEziRSc33p4RAsn5Q27dNR/image.png]. The Theorem 4I serve to characterize the binary operation + in a recursive fashion.
We now proceed to construct the multiplication operation in much the same way.
Definition Multiplication [IMAGE: https://cdn.steemitimages.com/DQmaXE3iZXqVpMmEXfyusvSXhuru4QZhUiXoeGBxMAQvux7/image.png] is the binary operation on [IMAGE: https://cdn.steemitimages.com/DQmdaaV7rUFiJseT7KCFst4Waxe1DjEusTyPoBwcuVS3jRb/image.png] such that for any m and n in [IMAGE: https://cdn.steemitimages.com/DQmdaaV7rUFiJseT7KCFst4Waxe1DjEusTyPoBwcuVS3jRb/image.png],
[IMAGE: https://cdn.steemitimages.com/DQmYZGpRGGwQK5SipjNGXWvKg21ibe8BrkDkxRvYPZffJvv/image.png]
This is just an extension of Theorem 4I.
Theorem 4J For natural numbers m and n,
[IMAGE: https://cdn.steemitimages.com/DQmfZ4G48cN8MT9pUZWBQkUrU4bWgRSJp1J6M9CF32uhgGk/image.png]
For convenience, we can now discard the [IMAGE: https://cdn.steemitimages.com/DQmdAKwKSZdry9obMKX8P4QYUAnT63JJ2GHm3kCSwkrXXvu/image.png] functions, and use [IMAGE: https://cdn.steemitimages.com/DQmXFB6K4NoLFhycFSCBcApxXNmWrtaCMVSQddz7aimVhYK/image.png] and use Theorem 4J instead.
We could, in the same manner, define the exponentiation operation on [IMAGE: https://cdn.steemitimages.com/DQmWb3wnYAyHyk7ZGynj8S8kYAGF36v8Y1MxgZxYRPZRKd4/image.png] The equations that characterize exponentiation are,
[IMAGE: https://cdn.steemitimages.com/DQmVw2xjFcvEVNbGyaC6GGiAK72mMCiKQmsUiWm7FwXYhT6/image.png]
Example Show 2 + 2 = 4
We can show the addition as follow:
[IMAGE: https://cdn.steemitimages.com/DQmPVoBUHu9ujDTs621QCMgHFoMYiVvPY8hqte49g5e16ao/image.png]
We have given some set-theoretic definitions of the operations of arithmetic, the next step is to verify that some of the common laws of arithmetic are provable within set theory.
Theorem 4K The following identities hold for all natural numbers.
(1) Associative law for addition
[IMAGE: https://cdn.steemitimages.com/DQmQ1Z8g4bjBTsWYDBNkBroQVnRWTJA5DZ8euUCfnmZq1wi/image.png]
(2) Commutative law for addition
[IMAGE: https://cdn.steemitimages.com/DQmXfm6QMX26Y2q1Y7pfbLqPoe9fx7KRENu2MRwjGA36aB4/image.png]
(3) Distributive law
[IMAGE: https://cdn.steemitimages.com/DQmZEEar8hjbf1R5CvKZZEZjYTtWUicMjSwYXH6mexy8EkR/image.png]
(4) Associative law for multiplication
[IMAGE: https://cdn.steemitimages.com/DQmYQLR8y8asGHveCPQLaRF49z4rmM1oVH7W8ZWBPTyMNtL/image.png]
(5) Commutative law for multiplication
[IMAGE: https://cdn.steemitimages.com/DQmNM56RUp39G1DryjXcKs1pT3vt9M4Jn1n71KYT1tKpEri/image.png]
Disclaimer: this is a summary of section 4.4 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshots from generated latex form using typora
Thank you for reading ...
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