Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: D`/l5%ja#*

modified

Before

Front

Consider the poset \((A; \preceq)\) and \( S \subseteq A\).

\(a \in A\) is a lower (upper) bound of \(S\) if \(b \preceq a\) (\(b \succeq a) \) for all \(b \in S\)

Back

Consider the poset \((A; \preceq)\) and \( S \subseteq A\).

\(a \in A\) is a lower (upper) bound of \(S\) if \(b \preceq a\) (\(b \succeq a) \) for all \(b \in S\)

Note that a is not necessarily in the subset S (difference to the least and greatest elements).

After

Front

Consider the poset \((A; \preceq)\) and \( S \subseteq A\).

\(a \in A\) is a lower (upper) bound of \(S\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in S\)

Back

Consider the poset \((A; \preceq)\) and \( S \subseteq A\).

\(a \in A\) is a lower (upper) bound of \(S\) if \(a \preceq b\) (\(a \succeq b) \) for all \(b \in S\)

Note that a is not necessarily in the subset S (difference to the least and greatest elements).

Field-by-field Comparison

Field	Before	After
Text	Consider the poset \((A; \preceq)\) and \( S \subseteq A\).<br><br><div>\(a \in A\) is a {{c1::<b>lower (upper) bound</b> of \(S\)}} if {{c2::\(b \preceq a\) (\(b \succeq a) \) for all \(b \in S\)}}</div>	Consider the poset \((A; \preceq)\) and \( S \subseteq A\).<br><br><div>\(a \in A\) is a {{c1::<b>lower (upper) bound</b> of \(S\)}} if {{c2::\(a \preceq b\) (\(a \succeq b) \) for all \(b \in S\)}}</div>

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::4._Special_Elements_in_Posets

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: I1&*hbv&c,

modified

Before

Front

An integral domain is a commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).

Back

An integral domain is a commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).

Examples: \(\mathbb{Z}, \mathbb{R}\)

Counterexample: \(\mathbb{Z}_m, m\) not prime

After

Front

An integral domain is a commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).

Back

An integral domain is a commutative ring without zerodivisors (\( \forall a \ \forall b \quad ab = 0 \rightarrow a = 0 \lor b = 0\) ).

A domain of elements behaving like integers.

Examples: \(\mathbb{Z}, \mathbb{R}\)

Counterexample: \(\mathbb{Z}_m, m\) not prime

Field-by-field Comparison

Field	Before	After
Extra	Examples: \(\mathbb{Z}, \mathbb{R}\)<div>Counterexample: \(\mathbb{Z}_m, m\) not prime</div>	<div><i>A domain of elements behaving like integers.</i></div><br>Examples: \(\mathbb{Z}, \mathbb{R}\)<div>Counterexample: \(\mathbb{Z}_m, m\) not prime</div>

Tags: ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::4._Zerodivisors_and_Integral_Domains

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Ni(5U1m?zz

modified

Before

Front

Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(\{ A \mid A \notin A\}\)}}.

Back

Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(\{ A \mid A \notin A\}\)}}.

After

Front

Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(R = \{ A \mid A \notin A \}\)}}.

Back

Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(R = \{ A \mid A \notin A \}\)}}.

Assume R contains itself

Then R should not contain itself (because R only contains sets that do not contain themselves).

➜ Contradiction.
Assume R does not contain itself

Then it does meet the rule for membership in R, so it should contain itself.

➜ Contradiction.

A barber that shaves all and only those men who do not shave themselves. Does he shave himself?

Field-by-field Comparison

Field	Before	After
Text	Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(\{ A \mid A \notin A\}\)}}.	Russell's Paradox proposes the (problematic) set \(R=\) {{c1::\(R = \{ A \mid A \notin A \}\)}}.
Extra		<div><ol><li> <div><b>Assume R contains itself</b><b></b></div> <div>Then R should <i>not</i> contain itself (because R only contains sets that do not contain themselves).</div> <div>➜ Contradiction.</div> </li><li> <div><b>Assume R does not contain itself</b><b></b></div> <div>Then it <i>does</i> meet the rule for membership in R, so it should contain itself.</div> <div>➜ Contradiction.</div></li></ol></div><i>A barber that shaves all and only those men who do not shave themselves. Does he shave himself?</i><br>

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::1._Introduction::2._Russell's_Paradox

Note 4: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: iYX;e6S}74

modified

Before

Front

The Hasse diagram of a poset \((A; \preceq)\) is the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).

Back

The Hasse diagram of a poset \((A; \preceq)\) is the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).

After

Front

The Hasse diagram of a poset \((A; \preceq)\) is defined as the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).

Back

The Hasse diagram of a poset \((A; \preceq)\) is defined as the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).

Field-by-field Comparison

Field	Before	After
Text	The <i>Hasse diagram</i> of a poset \((A; \preceq)\) is {{c1::the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).~~::defined as?~~}}	The <i>Hasse diagram</i> of a poset \((A; \preceq)\) is defined as {{c1::the directed graph whose vertices are the elements of \(A\) and where there is an edge from \(a\) to \(b\) if and only if \(b\) covers \(a\).}}

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::2._Hasse_Diagrams PlsFix::DUPLICATE