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Schröder, Ernst: Vorlesungen über die Algebra der Logik. Bd. 2, Abt. 1. Leipzig, 1891.

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§ 29. Übersicht der Sätze.

°35) Th. Satz vom Dualismus (vergl. § 14):
[Formel 1] .
Derselbe gehört eigentlich nicht zu den von unsrer Zeichensprache
beherrschten Sätzen, besitzt vielmehr seine besondre Symbolik.

°36) De Morgan's Theoreme:

°36x) (a b)1 = a1 + b1°36+) (a + b)1 = a1 b1,
oder a + b = (a1 b1)1.a b = (a1 + b1)1.

37) Th. der (Konversion durch) Kontraposition:
(a b) = (b1 a1).

38) Th. (a b1 = 0) = (a b) = (a1 + b = 1).

Zusatz.

(a b = 0) = (a b1) = (b a1), (a + b = 1) = (a1 b) = (b1 a).

39) Th. (a b1 + a1 b = 0) = (a = b) = (a b + a1 b1 = 1),
{(a + b) (a1 + b1) = 0} = (a = b) = {(a + b1) (a1 + b) = 1}

40) Th.(a c b c) (a + c b + c) = (a b)Schröder.
Zusatz 1.(a c = b c) (a + c = b + c) = (a = b),"
Zusatz 2.(a c b) (a b + c) = (a b),Peirce.

41) Th. von Peirce:
(a b c) = (a b1 + c) = (b a1 + c), (a b + c) = (a b1 c) = (a c1 b).

°42+) Th. y = (x y + u x1) x + (x1 y + v x) x1°42x) Th. Etc.

43) Th. [Formel 2] (a = u b) = (a b) = [Formel 3] (b = a + v).

°44+) Th. f (x) = f (1) x + f (0) x144x) f (x) = {f (0) + x} · {f (1) + x1}

Zusatz+. f (x, y) = f (1, 1) x y + f (1, 0) x y1 + f (0, 1) x1 y + f (0, 0) x1 y1,
Etc. (Boole und Peirce).

°Vorbemerkung zu Th. 45+):
(a x + b x1) + (a' x + b' x1) = (a + a') x + (b + b') x1,
(a x y + b x y1 + c x1 y + d x1 y1) + (a' x y + b' x y1 + c' x1 y + d' x1 y1) =
= (a + a') x y + (b + b') x y1 + (c + c') x1 y + (d + d') x1 y1, Etc.

°45+) Th. (a x + b x1) (a' x + b' x1) = a a' x + b b' x1,
(a x y + b x y1 + c x1 y + d x1) (a' x y + b' x y1 + c' x1 y + d' x1 y1) =
= a a' x y + b b' x y1 + c c' x1 y + d d' x1 y1, Etc. (Boole.)

Schröder, Algebra der Logik. II. 3
§ 29. Übersicht der Sätze.

°35) Th. Satz vom Dualismus (vergl. § 14):
[Formel 1] .
Derselbe gehört eigentlich nicht zu den von unsrer Zeichensprache
beherrschten Sätzen, besitzt vielmehr seine besondre Symbolik.

°36) De Morgan’s Theoreme:

°36×) (a b)1 = a1 + b1°36+) (a + b)1 = a1 b1,
oder a + b = (a1 b1)1.a b = (a1 + b1)1.

37) Th. der (Konversion durch) Kontraposition:
(a ⊆ b) = (b1 ⊆ a1).

38) Th. (a b1 = 0) = (a ⊆ b) = (a1 + b = 1).

Zusatz.

(a b = 0) = (a ⊆ b1) = (b ⊆ a1), (a + b = 1) = (a1 ⊆ b) = (b1 ⊆ a).

39) Th. (a b1 + a1 b = 0) = (a = b) = (a b + a1 b1 = 1),
{(a + b) (a1 + b1) = 0} = (a = b) = {(a + b1) (a1 + b) = 1}

40) Th.(a c ⊆ b c) (a + c ⊆ b + c) = (a ⊆ b)Schröder.
Zusatz 1.(a c = b c) (a + c = b + c) = (a = b),„
Zusatz 2.(a c ⊆ b) (a ⊆ b + c) = (a ⊆ b),Peirce.

41) Th. von Peirce:
(a b ⊆ c) = (a ⊆ b1 + c) = (b ⊆ a1 + c), (a ⊆ b + c) = (a b1 ⊆ c) = (a c1 ⊆ b).

°42+) Th. y = (x y + u x1) x + (x1 y + v x) x1°42×) Th. Etc.

43) Th. [Formel 2] (a = u b) = (a ⊆ b) = [Formel 3] (b = a + v).

°44+) Th. f (x) = f (1) x + f (0) x144×) f (x) = {f (0) + x} · {f (1) + x1}

Zusatz+. f (x, y) = f (1, 1) x y + f (1, 0) x y1 + f (0, 1) x1 y + f (0, 0) x1 y1,
Etc. (Boole und Peirce).

°Vorbemerkung zu Th. 45+):
(a x + b x1) + (a' x + b' x1) = (a + a') x + (b + b') x1,
(a x y + b x y1 + c x1 y + d x1 y1) + (a' x y + b' x y1 + c' x1 y + d' x1 y1) =
= (a + a') x y + (b + b') x y1 + (c + c') x1 y + (d + d') x1 y1, Etc.

°45+) Th. (a x + b x1) (a' x + b' x1) = a a' x + b b' x1,
(a x y + b x y1 + c x1 y + d x1) (a' x y + b' x y1 + c' x1 y + d' x1 y1) =
= a a' x y + b b' x y1 + c c' x1 y + d d' x1 y1, Etc. (Boole.)

Schröder, Algebra der Logik. II. 3
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            <fw place="top" type="header">§ 29. Übersicht der Sätze.</fw><lb/>
            <p>°35) <hi rendition="#g">Th. Satz vom Dualismus</hi> (vergl. § 14):<lb/><hi rendition="#c"><formula/>.</hi><lb/>
Derselbe gehört eigentlich nicht zu den von unsrer Zeichensprache<lb/>
beherrschten Sätzen, besitzt vielmehr seine besondre Symbolik.</p><lb/>
            <p>°36) <hi rendition="#g">De Morgan&#x2019;s Theoreme</hi>:<lb/><table><row><cell>°36<hi rendition="#sub">×</hi>) (<hi rendition="#i">a b</hi>)<hi rendition="#sub">1</hi> = <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi></cell><cell>°36<hi rendition="#sub">+</hi>) (<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>)<hi rendition="#sub">1</hi> = <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>,</cell></row><lb/><row><cell>oder <hi rendition="#i">a</hi> + <hi rendition="#i">b</hi> = (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>)<hi rendition="#sub">1</hi>.</cell><cell><hi rendition="#i">a b</hi> = (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>)<hi rendition="#sub">1</hi>.</cell></row><lb/></table></p>
            <p>37) <hi rendition="#g">Th. der</hi> (<hi rendition="#g">Konversion</hi> durch) <hi rendition="#g">Kontraposition</hi>:<lb/><hi rendition="#c">(<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>) = (<hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">a</hi><hi rendition="#sub">1</hi>).</hi></p><lb/>
            <p>38) <hi rendition="#g">Th.</hi> <hi rendition="#et">(<hi rendition="#i">a b</hi><hi rendition="#sub">1</hi> = 0) = (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>) = (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi> = 1).</hi></p><lb/>
            <p><hi rendition="#g">Zusatz</hi>.</p><lb/>
            <p>(<hi rendition="#i">a b</hi> = 0) = (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>) = (<hi rendition="#i">b</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">a</hi><hi rendition="#sub">1</hi>), (<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi> = 1) = (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>) = (<hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">a</hi>).</p><lb/>
            <p>39) <hi rendition="#g">Th.</hi> <hi rendition="#et">(<hi rendition="#i">a b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi> = 0) = (<hi rendition="#i">a</hi> = <hi rendition="#i">b</hi>) = (<hi rendition="#i">a b</hi> + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> = 1),<lb/>
{(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>) = 0} = (<hi rendition="#i">a</hi> = <hi rendition="#i">b</hi>) = {(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi>) = 1}</hi></p><lb/>
            <table>
              <row>
                <cell>40) <hi rendition="#g">Th.</hi></cell>
                <cell>(<hi rendition="#i">a c</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b c</hi>) (<hi rendition="#i">a</hi> + <hi rendition="#i">c</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) = (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>)</cell>
                <cell><hi rendition="#g">Schröder</hi>.</cell>
              </row><lb/>
              <row>
                <cell><hi rendition="#g">Zusatz</hi> 1.</cell>
                <cell>(<hi rendition="#i">a c</hi> = <hi rendition="#i">b c</hi>) (<hi rendition="#i">a</hi> + <hi rendition="#i">c</hi> = <hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) = (<hi rendition="#i">a</hi> = <hi rendition="#i">b</hi>),</cell>
                <cell>&#x201E;</cell>
              </row><lb/>
              <row>
                <cell><hi rendition="#g">Zusatz</hi> 2.</cell>
                <cell>(<hi rendition="#i">a c</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>) (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) = (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>),</cell>
                <cell><hi rendition="#g">Peirce</hi>.</cell>
              </row><lb/>
            </table>
            <p>41) <hi rendition="#g">Th.</hi> von <hi rendition="#g">Peirce</hi>:<lb/>
(<hi rendition="#i">a b</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">c</hi>) = (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi>) = (<hi rendition="#i">b</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi>), (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) = (<hi rendition="#i">a b</hi><hi rendition="#sub">1</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">c</hi>) = (<hi rendition="#i">a c</hi><hi rendition="#sub">1</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>).</p><lb/>
            <milestone rendition="#hr" unit="section"/>
            <table>
              <row>
                <cell>°42<hi rendition="#sub">+</hi>) <hi rendition="#g">Th.</hi> <hi rendition="#i">y</hi> = (<hi rendition="#i">x y</hi> + <hi rendition="#i">u x</hi><hi rendition="#sub">1</hi>) <hi rendition="#i">x</hi> + (<hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">v x</hi>) <hi rendition="#i">x</hi><hi rendition="#sub">1</hi></cell>
                <cell>°42<hi rendition="#sub">×</hi>) <hi rendition="#g">Th.</hi> Etc.</cell>
              </row><lb/>
            </table>
            <p>43) <hi rendition="#g">Th.</hi> <hi rendition="#et"><formula/> (<hi rendition="#i">a</hi> = <hi rendition="#i">u b</hi>) = (<hi rendition="#i">a</hi> <choice><orig>&#xFFFC;</orig><reg>&#x2286;</reg></choice> <hi rendition="#i">b</hi>) = <formula/> (<hi rendition="#i">b</hi> = <hi rendition="#i">a</hi> + <hi rendition="#i">v</hi>).</hi></p><lb/>
            <table>
              <row>
                <cell>°44<hi rendition="#sub">+</hi>) <hi rendition="#g">Th.</hi> <hi rendition="#i">f</hi> (<hi rendition="#i">x</hi>) = <hi rendition="#i">f</hi> (1) <hi rendition="#i">x</hi> + <hi rendition="#i">f</hi> (0) <hi rendition="#i">x</hi><hi rendition="#sub">1</hi></cell>
                <cell>44<hi rendition="#sub">×</hi>) <hi rendition="#i">f</hi> (<hi rendition="#i">x</hi>) = {<hi rendition="#i">f</hi> (0) + <hi rendition="#i">x</hi>} · {<hi rendition="#i">f</hi> (1) + <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>}</cell>
              </row><lb/>
            </table>
            <p><hi rendition="#g">Zusatz</hi><hi rendition="#sub">+</hi>. <hi rendition="#et"><hi rendition="#i">f</hi> (<hi rendition="#i">x</hi>, <hi rendition="#i">y</hi>) = <hi rendition="#i">f</hi> (1, 1) <hi rendition="#i">x y</hi> + <hi rendition="#i">f</hi> (1, 0) <hi rendition="#i">x y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">f</hi> (0, 1) <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">f</hi> (0, 0) <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>,<lb/>
Etc. (<hi rendition="#g">Boole</hi> und <hi rendition="#g">Peirce</hi>).</hi></p><lb/>
            <p>°<hi rendition="#g">Vorbemerkung zu Th</hi>. 45<hi rendition="#sub">+</hi>):<lb/><hi rendition="#c">(<hi rendition="#i">a x</hi> + <hi rendition="#i">b x</hi><hi rendition="#sub">1</hi>) + (<hi rendition="#i">a</hi>' <hi rendition="#i">x</hi> + <hi rendition="#i">b</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>) = (<hi rendition="#i">a</hi> + <hi rendition="#i">a</hi>') <hi rendition="#i">x</hi> + (<hi rendition="#i">b</hi> + <hi rendition="#i">b</hi>') <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>,<lb/>
(<hi rendition="#i">a x y</hi> + <hi rendition="#i">b x y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">d x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>) + (<hi rendition="#i">a</hi>' <hi rendition="#i">x y</hi> + <hi rendition="#i">b</hi>' <hi rendition="#i">x y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">d</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>) =<lb/>
= (<hi rendition="#i">a</hi> + <hi rendition="#i">a</hi>') <hi rendition="#i">x y</hi> + (<hi rendition="#i">b</hi> + <hi rendition="#i">b</hi>') <hi rendition="#i">x y</hi><hi rendition="#sub">1</hi> + (<hi rendition="#i">c</hi> + <hi rendition="#i">c</hi>') <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + (<hi rendition="#i">d</hi> + <hi rendition="#i">d</hi>') <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>, Etc.</hi></p><lb/>
            <p>°45<hi rendition="#sub">+</hi>) <hi rendition="#g">Th.</hi> <hi rendition="#et">(<hi rendition="#i">a x</hi> + <hi rendition="#i">b x</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">a</hi>' <hi rendition="#i">x</hi> + <hi rendition="#i">b</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>) = <hi rendition="#i">a a</hi>' <hi rendition="#i">x</hi> + <hi rendition="#i">b b</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>,</hi><lb/>
(<hi rendition="#i">a x y</hi> + <hi rendition="#i">b x y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">d x</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">a</hi>' <hi rendition="#i">x y</hi> + <hi rendition="#i">b</hi>' <hi rendition="#i">x y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">d</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>) =<lb/><hi rendition="#et">= <hi rendition="#i">a a</hi>' <hi rendition="#i">x y</hi> + <hi rendition="#i">b b</hi>' <hi rendition="#i">x y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c c</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi> + <hi rendition="#i">d d</hi>' <hi rendition="#i">x</hi><hi rendition="#sub">1</hi> <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>, Etc. (<hi rendition="#g">Boole</hi>.)</hi></p><lb/>
            <fw place="bottom" type="sig"><hi rendition="#k">Schröder</hi>, Algebra der Logik. II. 3</fw><lb/>
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[33/0057] § 29. Übersicht der Sätze. °35) Th. Satz vom Dualismus (vergl. § 14): [FORMEL]. Derselbe gehört eigentlich nicht zu den von unsrer Zeichensprache beherrschten Sätzen, besitzt vielmehr seine besondre Symbolik. °36) De Morgan’s Theoreme: °36×) (a b)1 = a1 + b1 °36+) (a + b)1 = a1 b1, oder a + b = (a1 b1)1. a b = (a1 + b1)1. 37) Th. der (Konversion durch) Kontraposition: (a  b) = (b1  a1). 38) Th. (a b1 = 0) = (a  b) = (a1 + b = 1). Zusatz. (a b = 0) = (a  b1) = (b  a1), (a + b = 1) = (a1  b) = (b1  a). 39) Th. (a b1 + a1 b = 0) = (a = b) = (a b + a1 b1 = 1), {(a + b) (a1 + b1) = 0} = (a = b) = {(a + b1) (a1 + b) = 1} 40) Th. (a c  b c) (a + c  b + c) = (a  b) Schröder. Zusatz 1. (a c = b c) (a + c = b + c) = (a = b), „ Zusatz 2. (a c  b) (a  b + c) = (a  b), Peirce. 41) Th. von Peirce: (a b  c) = (a  b1 + c) = (b  a1 + c), (a  b + c) = (a b1  c) = (a c1  b). °42+) Th. y = (x y + u x1) x + (x1 y + v x) x1 °42×) Th. Etc. 43) Th. [FORMEL] (a = u b) = (a  b) = [FORMEL] (b = a + v). °44+) Th. f (x) = f (1) x + f (0) x1 44×) f (x) = {f (0) + x} · {f (1) + x1} Zusatz+. f (x, y) = f (1, 1) x y + f (1, 0) x y1 + f (0, 1) x1 y + f (0, 0) x1 y1, Etc. (Boole und Peirce). °Vorbemerkung zu Th. 45+): (a x + b x1) + (a' x + b' x1) = (a + a') x + (b + b') x1, (a x y + b x y1 + c x1 y + d x1 y1) + (a' x y + b' x y1 + c' x1 y + d' x1 y1) = = (a + a') x y + (b + b') x y1 + (c + c') x1 y + (d + d') x1 y1, Etc. °45+) Th. (a x + b x1) (a' x + b' x1) = a a' x + b b' x1, (a x y + b x y1 + c x1 y + d x1) (a' x y + b' x y1 + c' x1 y + d' x1 y1) = = a a' x y + b b' x y1 + c c' x1 y + d d' x1 y1, Etc. (Boole.) Schröder, Algebra der Logik. II. 3

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Zitationshilfe: Schröder, Ernst: Vorlesungen über die Algebra der Logik. Bd. 2, Abt. 1. Leipzig, 1891, S. 33. In: Deutsches Textarchiv <https://www.deutschestextarchiv.de/schroeder_logik0201_1891/57>, abgerufen am 27.04.2024.