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integer [2013/09/03 00:40] nikolaj |
integer [2014/03/21 11:11] (current) |
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| ===== Integer ===== | ===== Integer ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#FFBB00: $ \mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\} $ | |
| with $a,b,n,m\in \mathbb N$. | with $a,b,n,m\in \mathbb N$. | ||
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| So if $[\langle a,b\rangle]$ is the equivalence class of $\langle a,b\rangle$ with respect to the equivalence relation $\{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\}$, we have | So if $[\langle a,b\rangle]$ is the equivalence class of $\langle a,b\rangle$ with respect to the equivalence relation $\{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\}$, we have | ||
| - | * $ 0 \equiv [\langle0,0\rangle] = [\langle1,1\rangle] = \dots = [\langlek,k\rangle] $ | + | * $ 0 \equiv [\langle0,0\rangle] = [\langle1,1\rangle] = \dots = [\langle k,k\rangle] $ |
| - | * $ 1 \equiv [\langle1,0\rangle] = [\langle2,1\rangle] = \dots = [\langlek+1,k\rangle] $ | + | * $ 1 \equiv [\langle1,0\rangle] = [\langle2,1\rangle] = \dots = [\langle k+1,k\rangle] $ |
| - | * $ -1 \equiv [\langle0,1\rangle] = [\langle1,2\rangle] = \dots = [\langlek,k+1\rangle] $ | + | * $ -1 \equiv [\langle0,1\rangle] = [\langle1,2\rangle] = \dots = [\langle k,k+1\rangle] $ |
| - | * $ 2 \equiv [\langle2,0\rangle] = [\langle3,1\rangle] = \dots = [\langlek+2,k\rangle] $ | + | * $ 2 \equiv [\langle2,0\rangle] = [\langle3,1\rangle] = \dots = [\langle k+2,k\rangle] $ |
| - | * $ -2 \equiv [\langle0,2\rangle] = [\langle1,3\rangle] = \dots = [\langlek,k+2\rangle] $ | + | * $ -2 \equiv [\langle0,2\rangle] = [\langle1,3\rangle] = \dots = [\langle k,k+2\rangle] $ |
| * $ 3 \equiv [\langle0,3\rangle] = \dots $ | * $ 3 \equiv [\langle0,3\rangle] = \dots $ | ||
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| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Integer|Integer]] | Wikipedia: [[http://en.wikipedia.org/wiki/Integer|Integer]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Subset of === | === Subset of === | ||
| [[Quotient set]] | [[Quotient set]] | ||
| === Refinement of === | === Refinement of === | ||
| [[Rational number]] | [[Rational number]] | ||
| - | === Requirements === | + | === Context === |
| [[Arithmetic structure of natural numbers]] | [[Arithmetic structure of natural numbers]] | ||
