MA0301/exam_template/.vscode/latex.code-snippets

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{
"exc": {
"scope": "latex",
"prefix": ["\\exc"],
"body":
[
"\\exc{}",
"\\begin{subexcs}",
" \\subexc{}",
" $0",
"\\end{subexcs}"
],
"description": "Adds new exc with subexcs",
},
"subexc": {
"scope": "latex",
"prefix": ["\\subexc"],
"body":
[
"\\subexc{}",
"\\begin{ssubexcs}",
" \\ssubexc{}",
" $0",
"\\end{ssubexcs}"
],
"description": "Adds new subexc with ssubexcs",
},
"graph-table": {
"scope": "latex",
"prefix": ["graph-table"],
"body":
[
"\\begin{figure}[H]",
" \\center",
" \\begin{tabular}{c|cc}",
" $A$ & \\multicolumn{2}{c}{$v$} \\\\",
" \\hline",
" & $a$ & $b$ \\\\",
" \\hline",
" $s_0$ & $s_$ & $s_$ \\\\",
" $s_1$ & $s_$ & $s_$ \\\\",
" $s_2$ & $s_$ & $s_$ \\\\",
" \\end{tabular}",
"\\end{figure}"
],
"description": "Adds graph-table",
},
"graph-table-double": {
"scope": "latex",
"prefix": ["graph-dtable"],
"body":
[
"\\begin{figure}[H]",
" \\center",
" \\begin{tabular}{c|cc}",
" $A$ & \\multicolumn{2}{c}{$v$} \\\\",
" \\hline",
" & $a$ & $b$ \\\\",
" \\hline",
" $s_0$ & $s_$ & $s_$ \\\\",
" $s_1$ & $s_$ & $s_$ \\\\",
" $s_2$ & $s_$ & $s_$ \\\\",
" \\end{tabular}",
"\\end{figure}"
],
"description": "Adds graph-table",
},
"graph-table-line": {
"scope": "latex",
"prefix": ["gtl"],
"body": " $s_$ & $s_$ & $s_$ \\\\",
"description": "Adds line inside graph-table",
},
"graph-table-double-line": {
"scope": "latex",
"prefix": ["gtdl"],
"body": " $s_$ & $s_$ & $s_$ & $s_$ & $s_$ \\\\",
"description": "Adds line inside a double graph-table",
},
"induction-proof": {
"scope": "latex",
"prefix": ["prove-induction"],
"body": [
"Base case:",
"",
"\\begin{align*}",
"",
"\\end{align*}",
"",
"Assume that",
"",
"\\[ $1 \\]",
"",
"Then",
"",
"\begin{align*}",
" &= $1 + \\\\",
" &=",
"\\end{align*}",
"",
"\\qed"
],
"description": "Template for induction proof",
},
"injective-proof": {
"scope": "latex",
"prefix": ["prove-injective"],
"body": [
"In order for $f(x)$ to be injective, it has to hold that",
"",
"\\[ f(a) = f(b) \\Rightarrow a = b \\]",
"",
"\\begin{align*}",
" f(a) &= f(b) \\\\",
" $0",
"\\end{align*}",
"",
"Hence $f(x)$ is injective."
],
"description": "Template for injective proof",
},
"surjective-proof": {
"scope": "latex",
"prefix": ["prove-surjective"],
"body": [
"In order for $f(x)$ to be surjective, it has to hold that",
"",
"\\[ \\forall x \\in SET \\exists y \\in SET [f(x) = y] \\]",
"",
"\\begin{align*}",
" y &= $1 \\\\",
"",
" x &= \\\\" ,
"\\end{align*}",
"",
"$ $ makes up all the elements in SET",
"",
"\\begin{align*}",
" f(y) &= \\\\",
"",
"\\end{align*}",
"",
"Hence $f(x)$ is surjective"
],
"description": "Template for surjective proof",
},
"bijective-proof": {
"scope": "latex",
"prefix": ["prove-bijective"],
"body": [
"\\textbf{Injective:}",
"",
"In order for $f(x)$ to be injective, it has to hold that",
"",
"\\[ f(a) = f(b) \\Rightarrow a = b \\]",
"",
"\\begin{align*}",
" f(a) &= f(b) \\\\",
" $0",
"\\end{align*}",
"",
"Hence $f(x)$ is injective.",
"",
"",
"\\textbf{Surjective:}",
"",
"In order for $f(x)$ to be surjective, it has to hold that",
"",
"\\[ \\forall x \\in SET \\exists y \\in SET [f(x) = y] \\]",
"",
"\\begin{align*}",
" y &= $1 \\\\",
"",
" x &= \\\\" ,
"\\end{align*}",
"",
"$ $ makes up all the elements in SET",
"",
"\\begin{align*}",
" f(y) &= \\\\",
"",
"\\end{align*}",
"",
"Hence $f(x)$ is surjective",
"",
"",
"\\textbf{Inverse:}",
"",
"The inverse is the same as the expression which makes up $x$ which we used to prove that $f(x)$ is surjective. Hence",
"",
"\\[ f^{-1}(x) = \\]",
],
"description": "Template for bijective proof",
},
"equivalence-relation-proof": {
"scope": "latex",
"prefix": ["prove-eq-rel"],
"body": [
"In order for this relation to be an equivalence equation, it has to be reflexive, symmetric and transitive.",
"",
"\\textbf{Reflexive:}",
"",
"\\[ \\]",
"",
"\\textbf{Symmetric:}",
"",
"\\[ \\]",
"",
"\\textbf{Transitive:}",
"",
"\\[ \\]",
"",
"Hence the relation is an equivalence relation",
],
"description": "Template for equivalence relation proof",
},
"partial order-proof": {
"scope": "latex",
"prefix": ["prove-poset"],
"body": [
"In order for this relation to be a partial order, it has to be reflexive, antisymmetric and transitive.",
"",
"\\textbf{Reflexive:}",
"",
"\\[ \\]",
"",
"\\textbf{Antisymmetric:}",
"",
"\\[ \\]",
"",
"\\textbf{Transitive:}",
"",
"\\[ \\]",
"",
"Hence the relation is a partial order",
],
"description": "Template for poset proof",
},
"poset minmax": {
"scope": "latex",
"prefix": ["minmax-poset"],
"body": [
"Minimal elements:",
" \\[ \\{ $0 \\} \\]",
"Maximal elements:",
" \\[ \\{ \\} \\]"
],
"description": "Minimal maximal elements for poset",
},
"Diagram": {
"scope": "latex",
"prefix": ["dia"],
"body": "\\includeDiagram[caption={}, width=1\\linewidth]{graphics/$0.tex}",
"description": "Include a diagram",
},
}