Elements of Set Theory: Ordered Pairs

Consider the following pair set: <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYchRPJgGqF2tFnft4ADttDxtFhqeWWSXpwfDDMoXjvky/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYchRPJgGqF2tFnft4ADttDxtFhqeWWSXpwfDDMoXjvky/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYchRPJgGqF2tFnft4ADttDxtFhqeWWSXpwfDDMoXjvky/image.png 2x" /> this can be thought as an unordered pair. Consider another pair set with additional information: <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSwEzZax6aKqD49p3zmELK1C5oYL58qsoMgPVnT3kFa13/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSwEzZax6aKqD49p3zmELK1C5oYL58qsoMgPVnT3kFa13/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSwEzZax6aKqD49p3zmELK1C5oYL58qsoMgPVnT3kFa13/image.png 2x" />, where 1 is the first component, 2 is the second component. What do we want from this? What we want is to define a set that uniquely encodes both what x and y are, and also what order they are in. In such a way that if I have this equation, <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmdm27bpcnyNyVa2RPurx8XZ1T91ANLKnQ1F85qkFwdi7y/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmdm27bpcnyNyVa2RPurx8XZ1T91ANLKnQ1F85qkFwdi7y/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmdm27bpcnyNyVa2RPurx8XZ1T91ANLKnQ1F85qkFwdi7y/image.png 2x" /> This would imply that <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSuMWC4s2bBwgo8heUmoXydmenxrdyHqmSFtXb2WSuEL8/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSuMWC4s2bBwgo8heUmoXydmenxrdyHqmSFtXb2WSuEL8/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSuMWC4s2bBwgo8heUmoXydmenxrdyHqmSFtXb2WSuEL8/image.png 2x" />. How would we state this definition? Let's consider first some definition that lacks our desired property. <ol> <li>If we define our ordered pair as <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbE88shxfnfNqyU71UFSUuYBnZ8nVZaTwbv85pFSQFtzr/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbE88shxfnfNqyU71UFSUuYBnZ8nVZaTwbv85pFSQFtzr/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmbE88shxfnfNqyU71UFSUuYBnZ8nVZaTwbv85pFSQFtzr/image.png 2x" />, then <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYiqsQ3qu5F6DcFp2wEnNw1FyJ6DT1bBvfyiirTEQu26n/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYiqsQ3qu5F6DcFp2wEnNw1FyJ6DT1bBvfyiirTEQu26n/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYiqsQ3qu5F6DcFp2wEnNw1FyJ6DT1bBvfyiirTEQu26n/image.png 2x" /> since <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVri4zJQjqbmwwMiHzipqxMfPoxH6jPVNAHPKckHb5R72/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVri4zJQjqbmwwMiHzipqxMfPoxH6jPVNAHPKckHb5R72/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVri4zJQjqbmwwMiHzipqxMfPoxH6jPVNAHPKckHb5R72/image.png 2x" />. So this is not the definition we're looking. <li>Consider this definition <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmayh3tuJehcQjXpMU9VpS9PaJ8EA9gyMogqL4FYekp8sQ/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmayh3tuJehcQjXpMU9VpS9PaJ8EA9gyMogqL4FYekp8sQ/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmayh3tuJehcQjXpMU9VpS9PaJ8EA9gyMogqL4FYekp8sQ/image.png 2x" />. Again the desired property fails, since <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZxZVGFBbfRizY2U4Eau4DTvbtbfVwiVQVfHzTKP8NCDr/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZxZVGFBbfRizY2U4Eau4DTvbtbfVwiVQVfHzTKP8NCDr/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmZxZVGFBbfRizY2U4Eau4DTvbtbfVwiVQVfHzTKP8NCDr/image.png 2x" />, both sides being equal to <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYHiHv3rPTLuwbwuXawrq8QRbdzP5mkngWQvUWURR6vqH/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYHiHv3rPTLuwbwuXawrq8QRbdzP5mkngWQvUWURR6vqH/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYHiHv3rPTLuwbwuXawrq8QRbdzP5mkngWQvUWURR6vqH/image.png 2x" />. The first successful definition of an ordered pair set was given by Norbert Wiener in 1914, who proposed to let, <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWKuYoUkeZEkVkVKhC94sBs6TfxSrssz9wbqanLooVCRq/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWKuYoUkeZEkVkVKhC94sBs6TfxSrssz9wbqanLooVCRq/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmWKuYoUkeZEkVkVKhC94sBs6TfxSrssz9wbqanLooVCRq/image.png 2x" /> A simpler definition was given by Kazimierz Kuratowski in 1921, and is the definition in general use today: <code>Definition <x,y> is defined to be <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmW7BCweFFNekTFb4kHWqncZ19WrkcN11iGku9G247fhN3/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmW7BCweFFNekTFb4kHWqncZ19WrkcN11iGku9G247fhN3/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmW7BCweFFNekTFb4kHWqncZ19WrkcN11iGku9G247fhN3/image.png 2x" />. We must prove that this definition captures the property that the ordered <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUJR4svn9K5J3QY1n414ybrucdDTNxNeXYRLoPGNzML33/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUJR4svn9K5J3QY1n414ybrucdDTNxNeXYRLoPGNzML33/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmUJR4svn9K5J3QY1n414ybrucdDTNxNeXYRLoPGNzML33/image.png 2x" /> uniquely determines both what x and y are, and the order upon them. <code>Theorem 3A <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNUAC8KafQLv51Crmo4N5AdfJGhcSpD4ZNXCTV6GQhVXM/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNUAC8KafQLv51Crmo4N5AdfJGhcSpD4ZNXCTV6GQhVXM/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmNUAC8KafQLv51Crmo4N5AdfJGhcSpD4ZNXCTV6GQhVXM/image.png 2x" /> iff <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeG5HdL3tohSYC7AVZvCC6hLkLV6L3cZHvZuCDp6XuD8k/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeG5HdL3tohSYC7AVZvCC6hLkLV6L3cZHvZuCDp6XuD8k/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmeG5HdL3tohSYC7AVZvCC6hLkLV6L3cZHvZuCDp6XuD8k/image.png 2x" />. Now suppose that we have two sets A and B, and we form ordered pairs <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZVsHBNcrELHfC97DptTqDhLu1wwtFpanjADFnaQwBbwV/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZVsHBNcrELHfC97DptTqDhLu1wwtFpanjADFnaQwBbwV/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmZVsHBNcrELHfC97DptTqDhLu1wwtFpanjADFnaQwBbwV/image.png 2x" /> with <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmehj8Y1L72eE1cUn8DCoZRSqPzZHoeFpYMR7KqY6avbYd/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmehj8Y1L72eE1cUn8DCoZRSqPzZHoeFpYMR7KqY6avbYd/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmehj8Y1L72eE1cUn8DCoZRSqPzZHoeFpYMR7KqY6avbYd/image.png 2x" /> and <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeYV4c1CgHqm23dYfzbKAJQ8PdLrSjr3zhxYFcuomHSsV/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeYV4c1CgHqm23dYfzbKAJQ8PdLrSjr3zhxYFcuomHSsV/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmeYV4c1CgHqm23dYfzbKAJQ8PdLrSjr3zhxYFcuomHSsV/image.png 2x" />. The collection of all such pairs is called the Cartesian product <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmckPuQT4J5U8KZQ62UureDtPQS3xvxrDUk3ZzwjjCctc1/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmckPuQT4J5U8KZQ62UureDtPQS3xvxrDUk3ZzwjjCctc1/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmckPuQT4J5U8KZQ62UureDtPQS3xvxrDUk3ZzwjjCctc1/image.png 2x" /> of A and B. <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWoFsZMNPZdCE84EY4GREBQ8bA6xkBYQEEM4NfqRLxMNi/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWoFsZMNPZdCE84EY4GREBQ8bA6xkBYQEEM4NfqRLxMNi/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmWoFsZMNPZdCE84EY4GREBQ8bA6xkBYQEEM4NfqRLxMNi/image.png 2x" /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVQuC6SQJxWhwbonnTKGhn2k5mXpw1oet5AMFxNDhtBAP/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVQuC6SQJxWhwbonnTKGhn2k5mXpw1oet5AMFxNDhtBAP/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVQuC6SQJxWhwbonnTKGhn2k5mXpw1oet5AMFxNDhtBAP/image.png 2x" /> The strategy to show that <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUxxxxV7pFZUMhC7pHnSiqnwVfRGqNQjEHPBnVXjFkxbB/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUxxxxV7pFZUMhC7pHnSiqnwVfRGqNQjEHPBnVXjFkxbB/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmUxxxxV7pFZUMhC7pHnSiqnwVfRGqNQjEHPBnVXjFkxbB/image.png 2x" /> is a set runs as follows. <ul> <li>If we can find a set that already contains all of the pairs<img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmScqxQoc2ZxxcMQra1xTjBVJQWkr3k3aWbyWCoNrJDbV9/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmScqxQoc2ZxxcMQra1xTjBVJQWkr3k3aWbyWCoNrJDbV9/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmScqxQoc2ZxxcMQra1xTjBVJQWkr3k3aWbyWCoNrJDbV9/image.png 2x" /> we want, then we can use a subset axiom to cut things down to . A suitable large set to start with is provided by the next lemma. <code>Lemma 3B If <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRJ47p1B6W7emhWypVSXC3AMUBksKZQ4G5EJHdK8vd2GN/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRJ47p1B6W7emhWypVSXC3AMUBksKZQ4G5EJHdK8vd2GN/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmRJ47p1B6W7emhWypVSXC3AMUBksKZQ4G5EJHdK8vd2GN/image.png 2x" /> and <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmPg5EL8YCzEkb45k4mEUpw8ok76Rpap73dbWCWpC4TXBP/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmPg5EL8YCzEkb45k4mEUpw8ok76Rpap73dbWCWpC4TXBP/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmPg5EL8YCzEkb45k4mEUpw8ok76Rpap73dbWCWpC4TXBP/image.png 2x" />, then <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNhMxB7YB8nMMppw5Dxq7wkQ9YXXA4vuoZpAXZ8Lc2yFi/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNhMxB7YB8nMMppw5Dxq7wkQ9YXXA4vuoZpAXZ8Lc2yFi/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmNhMxB7YB8nMMppw5Dxq7wkQ9YXXA4vuoZpAXZ8Lc2yFi/image.png 2x" /> <hr /> Proof: The fact that the braces <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVSBN2PiDMBhDHtVmZF5V3uMcaifXA6YYpJuX8wMXz3Sg/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVSBN2PiDMBhDHtVmZF5V3uMcaifXA6YYpJuX8wMXz3Sg/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVSBN2PiDMBhDHtVmZF5V3uMcaifXA6YYpJuX8wMXz3Sg/image.png 2x" /> are nested to a depth of 2 is responsible for the two applications of the power set operation; <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUmV4wBx2wxAN5NBxfAVBUdcWpGuPPy732BmJ8PCn92kt/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUmV4wBx2wxAN5NBxfAVBUdcWpGuPPy732BmJ8PCn92kt/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmUmV4wBx2wxAN5NBxfAVBUdcWpGuPPy732BmJ8PCn92kt/image.png 2x" /> Then we have, <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmU4iJQCXYjD8ShE5ReRUXWuvoLjxnrga7hPvTeGdxcLsw/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmU4iJQCXYjD8ShE5ReRUXWuvoLjxnrga7hPvTeGdxcLsw/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmU4iJQCXYjD8ShE5ReRUXWuvoLjxnrga7hPvTeGdxcLsw/image.png 2x" /> <hr /> <code>Corollary 3C For any sets A and B, there is a set whose members are exactly the pairs <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQQdWhkYbQ1SVL2qWWvhJoqExFUMkztss4dQ9aeRjZKSe/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQQdWhkYbQ1SVL2qWWvhJoqExFUMkztss4dQ9aeRjZKSe/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQQdWhkYbQ1SVL2qWWvhJoqExFUMkztss4dQ9aeRjZKSe/image.png 2x" /> with <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmcivgRcd6PamjR9rFwj9jDFNQe3cbQqfdw6QStizsf2KS/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmcivgRcd6PamjR9rFwj9jDFNQe3cbQqfdw6QStizsf2KS/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmcivgRcd6PamjR9rFwj9jDFNQe3cbQqfdw6QStizsf2KS/image.png 2x" />. <hr /> Proof: From a subset axiom we can construct <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmV95FPooe4CNkxCYgGi7zRCFkwZKPwNFqBjR4PUoQAXjf/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmV95FPooe4CNkxCYgGi7zRCFkwZKPwNFqBjR4PUoQAXjf/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmV95FPooe4CNkxCYgGi7zRCFkwZKPwNFqBjR4PUoQAXjf/image.png 2x" /> <hr /> This corollary justifies our earlier definition of the Cartesian product <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWhUaAGRifZ8y9Cy4BKLp7MGtbuEDJ5FsqueebcBWeLVx/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWhUaAGRifZ8y9Cy4BKLp7MGtbuEDJ5FsqueebcBWeLVx/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmWhUaAGRifZ8y9Cy4BKLp7MGtbuEDJ5FsqueebcBWeLVx/image.png 2x" />. <hr /> Disclaimer: this is a summary of section 3.1 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 screenshot from generated latex form using typora <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 <center> Thank you for reading ... <img src="https://images.hive.blog/0x0/https://steemitimages.com/0x0/https://media.giphy.com/media/ohdY5OaQmUmVW/giphy.gif" /> <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" />