Universal Instantiation:
Note 1: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
L[2O7285Lj
Before
Front
Back
Universal Instantiation:
For any formula \(F\) and any term \(t\) we have \[\forall x F \models F[x/t]\]
After
Front
What is the definition of universal instantiation?
Back
What is the definition of universal instantiation?
For any formula \(F\) and any term \(t\) we have: \[\forall x F \models F[x/t]\]
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <b> |
<b>What is the definition of universal instantiation?</b> |
| Back | For any formula \(F\) and any term \(t\) we have \[\forall x F \models F[x/t]\] | For any formula \(F\) and any term \(t\) we have: \[\forall x F \models F[x/t]\] |
Note 2: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID:
modified
Note Type: Horvath Cloze
GUID:
Oj3Xy8Rn2M
Before
Front
Skolem Normal Form has only universal quantifiers.
It is equisatisfiable (not equivalent!) to the original formula.
It is equisatisfiable (not equivalent!) to the original formula.
Back
Skolem Normal Form has only universal quantifiers.
It is equisatisfiable (not equivalent!) to the original formula.
It is equisatisfiable (not equivalent!) to the original formula.
After
Front
Skolem normal form has no existance quantifiers.
It is equisatisfiable (not equivalent!) to the original formula.
It is equisatisfiable (not equivalent!) to the original formula.
Back
Skolem normal form has no existance quantifiers.
It is equisatisfiable (not equivalent!) to the original formula.
It is equisatisfiable (not equivalent!) to the original formula.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Skolem |
Skolem normal form has {{c1::no existance quantifiers}}.<br>It is {{c2::<i>equisatisfiable</i> (not equivalent!)}} to the original formula. |
Note 3: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
kH8u]Z~QoA
Before
Front
Prenex form defintion:
Back
Prenex form defintion:
A formula of the form \[Q_1 x_1 \ Q_2 x_2 \ \dots \ Q_n x_n G\]where the \(Q_i\) are arbitrary quantifiers and \(G\) is a formula free of quantifiers.
After
Front
What is the definition of the prenex form?
Back
What is the definition of the prenex form?
A formula of the form \[Q_1 x_1 \ Q_2 x_2 \ \dots \ Q_n x_n G\]where the \(Q_i\) are arbitrary quantifiers and \(G\) is a formula free of quantifiers.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <b> |
<b>What is the definition of the prenex form?</b> |
Note 4: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
wJ,ON3lFCv
Before
Front
Give an example of an extension field constructed from polynomials.
Back
Give an example of an extension field constructed from polynomials.
\(\mathbb{R}[x]_{x^2+1}\) is a field, isomorphic to \(\mathbb{C}\) (the complex numbers).
Explanation: \(x^2 + 1\) is irreducible over \(\mathbb{R}\) (no real roots). Elements of \(\mathbb{R}[x]_{x^2+1}\) are of the form \(a + bx\) where \(x^2 = -1\), which corresponds exactly to complex numbers \(a + bi\).
There are no other extension fields on \(\mathbb{R}\) that aren't isomorphic to \(\mathbb{C}\).
After
Front
Give an example of an extension field constructed from polynomials.
Back
Give an example of an extension field constructed from polynomials.
\(\mathbb{R}[x]_{x^2+1}\) is a field, isomorphic to \(\mathbb{C}\) (the complex numbers).
Explanation: \(x^2 + 1\) is irreducible over \(\mathbb{R}\) (no real roots). Elements of \(\mathbb{R}[x]_{x^2+1}\) are of the form \(a + bx\) where \(x^2 = -1\), which corresponds exactly to complex numbers \(a + bi\).
Every proper finite extension field of \(\mathbb{R}\) is isomorphic to \(\mathbb{C}\).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Back | <p>\(\mathbb{R}[x]_{x^2+1}\) is a field, <strong>isomorphic to \(\mathbb{C}\)</strong> (the complex numbers).</p>
<p><strong>Explanation</strong>: \(x^2 + 1\) is irreducible over \(\mathbb{R}\) (no real roots). Elements of \(\mathbb{R}[x]_{x^2+1}\) are of the form \(a + bx\) where \(x^2 = -1\), which corresponds exactly to complex numbers \(a + bi\).</p>
<p> |
<p>\(\mathbb{R}[x]_{x^2+1}\) is a field, <strong>isomorphic to \(\mathbb{C}\)</strong> (the complex numbers).</p> <p><strong>Explanation</strong>: \(x^2 + 1\) is irreducible over \(\mathbb{R}\) (no real roots). Elements of \(\mathbb{R}[x]_{x^2+1}\) are of the form \(a + bx\) where \(x^2 = -1\), which corresponds exactly to complex numbers \(a + bi\).</p> <p>Every proper finite extension field of \(\mathbb{R}\) is isomorphic to \(\mathbb{C}\).</p> |