Elements of Set Theory: Ordering on Natural Numbers

in #mathematics6 years ago

Ordering on Natural Numbers

<p dir="auto">Previously, we have defined natural numbers that was a side effect of our spurious definition, e.g. . Also, we have the following simple definition of order on natural numbers:<a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYntBag1mJjacnKgs2L4Uu7VcDz2H5mBga4jeZLiHogbA/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYntBag1mJjacnKgs2L4Uu7VcDz2H5mBga4jeZLiHogbA/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYntBag1mJjacnKgs2L4Uu7VcDz2H5mBga4jeZLiHogbA/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYntBag1mJjacnKgs2L4Uu7VcDz2H5mBga4jeZLiHogbA/image.png 2x" /> <blockquote> <p dir="auto">For natural numbers <em>m and <em>n, define <em>m to be less than <em>n iff <a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSRPpeiK9cbZGgjgXytQdZ7hAHR3KyxtT9F8KQMFDGYB6/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSRPpeiK9cbZGgjgXytQdZ7hAHR3KyxtT9F8KQMFDGYB6/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSRPpeiK9cbZGgjgXytQdZ7hAHR3KyxtT9F8KQMFDGYB6/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSRPpeiK9cbZGgjgXytQdZ7hAHR3KyxtT9F8KQMFDGYB6/image.png 2x" /> <p dir="auto">We then, introduce a special symbol "<" for this:<br /> <center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmT147i6N2jC9gnS1dPC4GegSnGR14CX9yDgbYJ532QuWP/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT147i6N2jC9gnS1dPC4GegSnGR14CX9yDgbYJ532QuWP/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT147i6N2jC9gnS1dPC4GegSnGR14CX9yDgbYJ532QuWP/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmT147i6N2jC9gnS1dPC4GegSnGR14CX9yDgbYJ532QuWP/image.png 2x" /> <p dir="auto">Note, this may seem redundant in a way that we can just use . But we have to keep in mind that the symbol "<" has a dual role which is to denote both membership and ordering.<a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSysEnX69kXW5MxvcEYbJ7R9QH8XLWE67UGEarpr35QHE/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSysEnX69kXW5MxvcEYbJ7R9QH8XLWE67UGEarpr35QHE/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSysEnX69kXW5MxvcEYbJ7R9QH8XLWE67UGEarpr35QHE/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSysEnX69kXW5MxvcEYbJ7R9QH8XLWE67UGEarpr35QHE/image.png 2x" /> <p dir="auto">Now, in place of an symbol, we define<br /> <center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQbRXV9ytTbJFJ7zhQfRHm6UZNyWBRQ2543zkcDSzVMt3/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQbRXV9ytTbJFJ7zhQfRHm6UZNyWBRQ2543zkcDSzVMt3/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQbRXV9ytTbJFJ7zhQfRHm6UZNyWBRQ2543zkcDSzVMt3/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQbRXV9ytTbJFJ7zhQfRHm6UZNyWBRQ2543zkcDSzVMt3/image.png 2x" /> <a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVGRiY9qCBQr297txUZUnASpDnQuJGW3PD7QtzSNuDfe1/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVGRiY9qCBQr297txUZUnASpDnQuJGW3PD7QtzSNuDfe1/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVGRiY9qCBQr297txUZUnASpDnQuJGW3PD7QtzSNuDfe1/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVGRiY9qCBQr297txUZUnASpDnQuJGW3PD7QtzSNuDfe1/image.png 2x" /> <p dir="auto">Observe that<br /> <center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmekEvoUL2jZStqin41MkPbucMv7W5USHgg2eny8DJ3kxg/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmekEvoUL2jZStqin41MkPbucMv7W5USHgg2eny8DJ3kxg/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmekEvoUL2jZStqin41MkPbucMv7W5USHgg2eny8DJ3kxg/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmekEvoUL2jZStqin41MkPbucMv7W5USHgg2eny8DJ3kxg/image.png 2x" /> <p dir="auto">a fact we will use in later calculations. <p dir="auto">We are now entitled to state the following fact: <blockquote> <p dir="auto">any natural number is just the set of all smaller natural numbers. <p dir="auto"><center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmd4789LvX5WJ5FC8qNsMusRdbiSnv4nD6rdoNxAuwYQSS/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmd4789LvX5WJ5FC8qNsMusRdbiSnv4nD6rdoNxAuwYQSS/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmd4789LvX5WJ5FC8qNsMusRdbiSnv4nD6rdoNxAuwYQSS/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmd4789LvX5WJ5FC8qNsMusRdbiSnv4nD6rdoNxAuwYQSS/image.png 2x" /> <p dir="auto">which is true because is a transitive set, and thus <a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmTK6bwp6msi1TTf9LU9w5djGzDSR1UUBiZjS6KbosUYBX/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTK6bwp6msi1TTf9LU9w5djGzDSR1UUBiZjS6KbosUYBX/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTK6bwp6msi1TTf9LU9w5djGzDSR1UUBiZjS6KbosUYBX/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmTK6bwp6msi1TTf9LU9w5djGzDSR1UUBiZjS6KbosUYBX/image.png 2x" /> <a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmT85rAWk3DxR7noCKCi8sPKASa4jkULgSsenx4Nqph7FF/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT85rAWk3DxR7noCKCi8sPKASa4jkULgSsenx4Nqph7FF/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT85rAWk3DxR7noCKCi8sPKASa4jkULgSsenx4Nqph7FF/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmT85rAWk3DxR7noCKCi8sPKASa4jkULgSsenx4Nqph7FF/image.png 2x" /> <p dir="auto">Because each natural number is a transitive set, we have for <em>m,n,p in <br /> <center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmeqfcN9X3mV4RQXEPuFrXwAcR6TofN9Lvqipih9DXZfzL/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeqfcN9X3mV4RQXEPuFrXwAcR6TofN9Lvqipih9DXZfzL/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeqfcN9X3mV4RQXEPuFrXwAcR6TofN9Lvqipih9DXZfzL/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmeqfcN9X3mV4RQXEPuFrXwAcR6TofN9Lvqipih9DXZfzL/image.png 2x" /> <a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmRrMGGUykMX2M7afAD7BmgygyvJbf4fywTYnphcPr52vo/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRrMGGUykMX2M7afAD7BmgygyvJbf4fywTYnphcPr52vo/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRrMGGUykMX2M7afAD7BmgygyvJbf4fywTYnphcPr52vo/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmRrMGGUykMX2M7afAD7BmgygyvJbf4fywTYnphcPr52vo/image.png 2x" /> <p dir="auto">That is, our ordering relation on is a transitive relation.<a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmd8z5LSTcmAqvg8bwqqFQiwXUUasPyV7XGJfANqScKWkL/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmd8z5LSTcmAqvg8bwqqFQiwXUUasPyV7XGJfANqScKWkL/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmd8z5LSTcmAqvg8bwqqFQiwXUUasPyV7XGJfANqScKWkL/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmd8z5LSTcmAqvg8bwqqFQiwXUUasPyV7XGJfANqScKWkL/image.png 2x" /> <p dir="auto">With this definition, it is hard to show that between two distinct natural numbers, one is larger than the other. For that, we need the following lemma, <p dir="auto"><code>Lemma 4L (a) For any natural numbers <em>m and <em>n,<br /> <center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmaWoc3RdyKntUpZJMjMHeGXWwWZbX3GL25GzWXPwPbbug/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmaWoc3RdyKntUpZJMjMHeGXWwWZbX3GL25GzWXPwPbbug/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmaWoc3RdyKntUpZJMjMHeGXWwWZbX3GL25GzWXPwPbbug/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmaWoc3RdyKntUpZJMjMHeGXWwWZbX3GL25GzWXPwPbbug/image.png 2x" /> <p dir="auto">(b) No natural number is a member of itself <p dir="auto">We next use the lemma to prove that for two distinct natural numbers, one is always a member of the other. <p dir="auto"><code>Trichotomy Law for set of natural numbers: For any natural numbers <em>m and <em>n, exactly one of the three conditions,<br /> <center><a href="https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQRpABBSiG6FGYvvxWYfsUmLHBUizGmUjkXUc9rXTjyZz/image.png"> <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQRpABBSiG6FGYvvxWYfsUmLHBUizGmUjkXUc9rXTjyZz/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQRpABBSiG6FGYvvxWYfsUmLHBUizGmUjkXUc9rXTjyZz/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQRpABBSiG6FGYvvxWYfsUmLHBUizGmUjkXUc9rXTjyZz/image.png 2x" /> <p dir="auto">holds. <p dir="auto"><br /> <hr /> <sub>Disclaimer: this is a summary of section 4.5 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 <br /> <br /> <ol> <li><a href="https://github.com/valjen/book_collection" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Elements of Set Theory by Herbert B. Enderton <p dir="auto"><center>Thank you for reading ... <p dir="auto"><br /><br /> <span><a href="https://images.hive.blog/0x0/https://steemitimages.com/0x0/https://media.giphy.com/media/ohdY5OaQmUmVW/giphy.gif"> <img src="https://images.hive.blog/0x0/https://steemitimages.com/0x0/https://media.giphy.com/media/ohdY5OaQmUmVW/giphy.gif" /> <span> <a href="https://images.hive.blog/1536x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png"> <img src="https://images.hive.blog/768x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png" srcset="https://images.hive.blog/768x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png 1x, https://images.hive.blog/1536x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png 2x" />
Sort:  

This post has received a 4.42 % upvote from @boomerang.

Congratulations @sinbad989! You have completed the following achievement on Steemit and have been rewarded with new badge(s) :

<p dir="auto"><a href="http://steemitboard.com/@sinbad989" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"><img src="https://images.hive.blog/768x0/https://steemitimages.com/70x80/http://steemitboard.com/notifications/votes.png" srcset="https://images.hive.blog/768x0/https://steemitimages.com/70x80/http://steemitboard.com/notifications/votes.png 1x, https://images.hive.blog/1536x0/https://steemitimages.com/70x80/http://steemitboard.com/notifications/votes.png 2x" /> Award for the number of upvotes <p dir="auto"><sub><em>Click on the badge to view your Board of Honor.<br /> <sub><em>If you no longer want to receive notifications, reply to this comment with the word <code>STOP <p dir="auto"><strong><span>Do not miss the last post from <a href="/@steemitboard">@steemitboard:<br /> <a href="https://steemit.com/veterans/@steemitboard/steemitboard-and-the-veterans-on-steemit-the-first-community-badge" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">SteemitBoard and the Veterans on Steemit - The First Community Badge. <blockquote> <p dir="auto">Do you like <a href="https://steemit.com/@steemitboard" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">SteemitBoard's project? Then <strong><a href="https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Vote for its witness and <strong>get one more award!


Congratulations @sinbad989!
You raised your level and are now a Minnow!

Do not miss the last post from @steemitboard:

Presentamos el Ranking de SteemitBoard
Introducing SteemitBoard Ranking

Support SteemitBoard's project! Vote for its witness and get one more award!

Congratulations @sinbad989! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

<p dir="auto"><a href="http://steemitboard.com/@sinbad989" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"><img src="https://images.hive.blog/768x0/https://steemitimages.com/70x80/http://steemitboard.com/notifications/votes.png" srcset="https://images.hive.blog/768x0/https://steemitimages.com/70x80/http://steemitboard.com/notifications/votes.png 1x, https://images.hive.blog/1536x0/https://steemitimages.com/70x80/http://steemitboard.com/notifications/votes.png 2x" /> Award for the number of upvotes <p dir="auto"><sub><em>Click on the badge to view your Board of Honor.<br /> <sub><em>If you no longer want to receive notifications, reply to this comment with the word <code>STOP <p dir="auto"><strong><span>Do not miss the last post from <a href="/@steemitboard">@steemitboard: <table><tr><td><a href="https://steemit.com/steemitboard/@steemitboard/steemitboard-ranking-update-resteem-and-resteemed-added" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"><img src="https://images.hive.blog/768x0/https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmfRVpHQhLDhnjDtqck8GPv9NPvNKPfMsDaAFDE1D9Er2Z/header_ranking.png" srcset="https://images.hive.blog/768x0/https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmfRVpHQhLDhnjDtqck8GPv9NPvNKPfMsDaAFDE1D9Er2Z/header_ranking.png 1x, https://images.hive.blog/1536x0/https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmfRVpHQhLDhnjDtqck8GPv9NPvNKPfMsDaAFDE1D9Er2Z/header_ranking.png 2x" /><td><a href="https://steemit.com/steemitboard/@steemitboard/steemitboard-ranking-update-resteem-and-resteemed-added" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">SteemitBoard Ranking update - Resteem and Resteemed added <blockquote> <p dir="auto">Support <a href="https://steemit.com/@steemitboard" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">SteemitBoard's project! <strong><a href="https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Vote for its witness and <strong>get one more award!

Congratulations @sinbad989! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

<table><tr><td><span><img src="https://images.hive.blog/768x0/https://steemitimages.com/60x70/http://steemitboard.com/@sinbad989/votes.png?201811082135" srcset="https://images.hive.blog/768x0/https://steemitimages.com/60x70/http://steemitboard.com/@sinbad989/votes.png?201811082135 1x, https://images.hive.blog/1536x0/https://steemitimages.com/60x70/http://steemitboard.com/@sinbad989/votes.png?201811082135 2x" /><td>You made more than 2000 upvotes. Your next target is to reach 3000 upvotes. <p dir="auto"><sub><em><a href="https://steemitboard.com/@sinbad989" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Click here to view your Board of Honor<br /> <sub><em>If you no longer want to receive notifications, reply to this comment with the word <code>STOP <p dir="auto"><strong><span>Do not miss the last post from <a href="/@steemitboard">@steemitboard: <table><tr><td><a href="https://steemit.com/steemfest/@steemitboard/steemfest3-and-steemitboard-meet-the-steemians-contest" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"><img src="https://images.hive.blog/768x0/https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmeLukvNFRsa7RURqsFpiLGEZZD49MiU52JtWmjS5S2wtW/image.png" srcset="https://images.hive.blog/768x0/https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmeLukvNFRsa7RURqsFpiLGEZZD49MiU52JtWmjS5S2wtW/image.png 1x, https://images.hive.blog/1536x0/https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmeLukvNFRsa7RURqsFpiLGEZZD49MiU52JtWmjS5S2wtW/image.png 2x" /><td><a href="https://steemit.com/steemfest/@steemitboard/steemfest3-and-steemitboard-meet-the-steemians-contest" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">SteemFest3 and SteemitBoard - Meet the Steemians Contest <blockquote> <p dir="auto">Support <a href="https://steemit.com/@steemitboard" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">SteemitBoard's project! <strong><a href="https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1" target="_blank" rel="noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Vote for its witness and <strong>get one more award!

Congratulations @sinbad989! You received a personal award!

1 Year on Steemit

Click here to view your Board of Honor

Support SteemitBoard's project! Vote for its witness and get one more award!

@tipu profit 1000 sp delegated

Yesterday 1000.0 STEEM POWER delegated or invested gave payout of:
0.042 SBD + 0.326 STEEM (0.15 USD), APR: 16.3% .

Delegation link: steemconnect 1000.0 SP delegation to @tipu.

Please note that your profit can be slightly different (depending on the payout time).

Check out https://www.steemprofit.info to compare @tipU with other services.

@tipu profit 1000 sp delegated

Yesterday 1000.0 STEEM POWER delegated or invested gave payout of:
0.095 SBD + 0.279 STEEM (0.23 USD), APR: 17.62% .

Delegation link: steemconnect 1000.0 SP delegation to @tipu.

Please note that your profit can be slightly different (depending on the payout time).

Check out https://www.steemprofit.info to compare @tipU with other services.