What is the number of subgroups of \(\mathbb{Z}_n\)?
Note 1: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
M035/^ZEJ$
Before
Front
Back
What is the number of subgroups of \(\mathbb{Z}_n\)?
it is the number of divisors of \(n\) (as the order of each subgroup divides the group order (which is n here) by Lagrange).
if \(n\) is written \(n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\)
Note: This only holds because \(\mathbb{Z}_n\) is cyclic and therefore the subgroups are unique.
if \(n\) is written \(n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\)
Note: This only holds because \(\mathbb{Z}_n\) is cyclic and therefore the subgroups are unique.
After
Front
What is the number of subgroups of \(\mathbb{Z}_n\)?
Back
What is the number of subgroups of \(\mathbb{Z}_n\)?
it is the number of divisors of \(n\) (as the order of each subgroup divides the group order (which is n here) by Lagrange).
if \(n\) is written \(n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\)
Note: This only holds because \(\mathbb{Z}_n\) is cyclic and therefore the subgroups are unique.
if \(n\) is written \(n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\)
Note: This only holds because \(\mathbb{Z}_n\) is cyclic and therefore the subgroups are unique.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Back | it is the number of divisors of \(n\) (as the order of each subgroup divides the group order (which is n here) by Lagrange). <br>if \(n\) is written \(n = p_1^{e_1} \ |
it is the number of divisors of \(n\) (as the order of each subgroup divides the group order (which is n here) by Lagrange). <br>if \(n\) is written \(n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}\) then it is \(\prod_{i=1}^k (e_i+1)\)<br><br><i>Note:</i> This only holds because \(\mathbb{Z}_n\) is cyclic and therefore the subgroups are unique. |
Note 2: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID:
modified
Note Type: Horvath Cloze
GUID:
Oj6Zv8Tn2M
Before
Front
In predicate logic interpretation, \(\varphi\) assigns {{c2::predicate symbols \(P\) to functions, \(\varphi(P)\) is a function \(U^k \rightarrow \{0,1\}\)}}.
Back
In predicate logic interpretation, \(\varphi\) assigns {{c2::predicate symbols \(P\) to functions, \(\varphi(P)\) is a function \(U^k \rightarrow \{0,1\}\)}}.
After
Front
In predicate logic interpretation, \(\psi\) assigns {{c2::predicate symbols \(P\) to functions, \(\psi(P)\) is a function \(U^k \rightarrow \{0,1\}\)}}.
Back
In predicate logic interpretation, \(\psi\) assigns {{c2::predicate symbols \(P\) to functions, \(\psi(P)\) is a function \(U^k \rightarrow \{0,1\}\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | In predicate logic interpretation, {{c1::\(\ |
In predicate logic interpretation, {{c1::\(\psi\)}} assigns {{c2::<b>predicate</b> symbols \(P\) to functions, \(\psi(P)\) is a function \(U^k \rightarrow \{0,1\}\)}}. |
Note 3: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID:
modified
Note Type: Horvath Cloze
GUID:
Rl8Ww2Tn9P
Before
Front
A set \(M\) of formulas is unsatisfiable if and only if {{c2::\(\mathcal{K}(M) \vdash_{\textbf{Res}} \emptyset\)}}.
Back
A set \(M\) of formulas is unsatisfiable if and only if {{c2::\(\mathcal{K}(M) \vdash_{\textbf{Res}} \emptyset\)}}.
After
Front
A set \(M\) of formulas is unsatisfiable if and only if {{c2::\(\mathcal{K}(M) \vdash_{Res} \emptyset\)}}.
Back
A set \(M\) of formulas is unsatisfiable if and only if {{c2::\(\mathcal{K}(M) \vdash_{Res} \emptyset\)}}.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | A set \(M\) of formulas is {{c1::unsatisfiable}} if and only if {{c2::\(\mathcal{K}(M) \vdash_{ |
A set \(M\) of formulas is {{c1::unsatisfiable}} if and only if {{c2::\(\mathcal{K}(M) \vdash_{Res} \emptyset\)}}. |