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Fungrim entry: c84f3f

SL2(Z)={(abcd):aZandbZandcZanddZandadbc=1}\operatorname{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \right\}
TeX:
\operatorname{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \right\}
Definitions:
Fungrim symbol Notation Short description
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("c84f3f"),
    Formula(Equal(SL2Z, SetBuilder(Matrix2x2(a, b, c, d), Tuple(a, b, c, d), And(Element(a, ZZ), Element(b, ZZ), Element(c, ZZ), Element(d, ZZ), Equal(Sub(Mul(a, d), Mul(b, c)), 1))))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC