Schröder, Ernst: Vorlesungen über die Algebra der Logik. Bd. 1. Leipzig, 1890.

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Neunte Vorlesung.

desgleichen unsymmetrisch, aber einfacher, als:
(a + b c) (b + a c) = a (b + c) + b (a + c), etc.
-- indem diese Ausdrücke alle durch Ausmultipliziren, nach dem Ab-
sorptionsgesetze auf a b + a c + b c hinauskommen. --
kh) (a + b c) b = (a + c) b; a (a b + b c) = a b;
(a b + a c + b c) a b c = a b c; (b + a c) (c + d) = a c + b c + b d;
a + b (c + d) + (a + b x) c = a + b (c + d); (a + b) (b + a c) = b + a c;
(a + b) (b + a) = a + b; (a + b) (b + c) (c + d) (d + a) = a c + b d;
(a + b) (b + c) = b + a c; (a + b) (b + c) (c + d) = a c + c b + b d;
(a + b) (b + c) (c + d) (d + e) = b d + c (a d + a e + b e);
(a + b) (b + c) (c + d) (d + e) (e + f) = a c d f + a c e + b c e + b d e + b d f;
(a + b) (a + c) (a + d) (b + c) (b + d) (c + d) = a b c + a b d + a c d + b c d;
(a + b) (b + c) (c + d) (d + e) (e + a) = a d b + b e c + c a d + d b e + e c a;
(a + b + c) (a + b + d) (a + c + d) (b + c + d) = a b + a c + a d + b c + b d + c d;
a (b + c) c (a + b) = a c;
(a + b c) (b + a c) = a b + a c + b c = (a + b) (a b + a c + b c);
(a b + c d) (a + b) (c + d) = a b (c + d) + (a + b) c d.

a (b + c) + c1 = a + c1; a (b + c) (a1 + b1) c1 = 0; a1 b c (b1 + a c + a1 c1) = 0;
(a + b) (a1 + b1) = a b1 + a1 b, (a + b1) (a1 + b) = a b + a1 b1;
(a + b1 c) (b + a1 c1) = a b, (a1 x + b) (a + b1 x1) = a b; a (b + c) + a1 + b1 c1 = 1;
(a + b1 c1) (a c + b1) = a c + b1 c1; a (b1 + c d) b (c1 + d) = a b c d;
(a b1 c + a1 b c1) (a b c1 + a1 b1 c) = 0; (a1 + b c) (a + b1 c1) = a b c + a1 b1 c1;
(a1 + b1) (a b + a c + b c) = c (a b1 + a1 b); (a1 + b1) a (b c1 + b1 c) = a b1 c;
a (b + c) (c1 + a b + a1 b1) = a b; a + b1 + c1 + b (a c1 + a1 c) = 1;
{a b1 c + (a1 + c1) b} {a b1 c1 + a1 (b + c)} = a1 b; a b + a b1 c = a (b + c);
a (a1 + b1 c) (a1 + b c) = a c (a1 + b1) (a1 + b) = a (b1 + a1 c) b (a1 + c1) = 0;
a1 (b1 + c1) (b + c a) (c + a b) = 0; (x + y) (x1 + y z1) (y1 + x z) = 0;
x y (a + x1) (a1 + y1) (b + y1) (b1 + x1) = 0; a1 + b1 + c1 + a b + a c + b c = 1;
(x + y) (x1 + z1) (y1 + z) (x1 + y + z) (x + y1 + z1) = 0; a1 + b1 + c (a b1 + a1 b) = a1 + b1;
(a + b) (a b1 + a1 b) = a b1 + a1 b = (a1 + b1) (a b1 + a1 b);
a b (a b1 + a1 b) = 0 = a1 b1 (a b1 + a1 b);
(b c1 + b1 c) (c a1 + c1 a) (a b1 + a1 b) = 0; (b c1 + b1 c) + (c a1 + c1 a) + (a b1 + a1 b) =
= (c a1 + c1 a) + (a b1 + a1 b) = (a b1 + a1 b) + (b c1 + b1 c) = (b c1 + b1 c) + (c a1 + c1 a) =
= (a + b + c) (a1 + b1 + c1) = a (b1 + c1) + a1 (b + c) = etc.

Neunte Vorlesung.

desgleichen unsymmetrisch, aber einfacher, als:
(a + b c) (b + a c) = a (b + c) + b (a + c), etc.
— indem diese Ausdrücke alle durch Ausmultipliziren, nach dem Ab-
sorptionsgesetze auf a b + a c + b c hinauskommen. —
χ) (a + b c) b = (a + c) b; a (a b + b c) = a b;
(a b + a c + b c) a b c = a b c; (b + a c) (c + d) = a c + b c + b d;
a + b (c + d) + (a + b x) c = a + b (c + d); (a + b) (b + a c) = b + a c;
(a + b) (b + a) = a + b; (a + b) (b + c) (c + d) (d + a) = a c + b d;
(a + b) (b + c) = b + a c; (a + b) (b + c) (c + d) = a c + c b + b d;
(a + b) (b + c) (c + d) (d + e) = b d + c (a d + a e + b e);
(a + b) (b + c) (c + d) (d + e) (e + f) = a c d f + a c e + b c e + b d e + b d f;
(a + b) (a + c) (a + d) (b + c) (b + d) (c + d) = a b c + a b d + a c d + b c d;
(a + b) (b + c) (c + d) (d + e) (e + a) = a d b + b e c + c a d + d b e + e c a;
(a + b + c) (a + b + d) (a + c + d) (b + c + d) = a b + a c + a d + b c + b d + c d;
a (b + c) c (a + b) = a c;
(a + b c) (b + a c) = a b + a c + b c = (a + b) (a b + a c + b c);
(a b + c d) (a + b) (c + d) = a b (c + d) + (a + b) c d.

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          <p><pb facs="#f0404" n="384"/><fw place="top" type="header">Neunte Vorlesung.</fw><lb/>
desgleichen unsymmetrisch, aber einfacher, als:<lb/><hi rendition="#c">(<hi rendition="#i">a</hi> + <hi rendition="#i">b c</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">a c</hi>) = <hi rendition="#i">a</hi> (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) + <hi rendition="#i">b</hi> (<hi rendition="#i">a</hi> + <hi rendition="#i">c</hi>), etc.</hi><lb/>
&#x2014; indem diese Ausdrücke alle durch Ausmultipliziren, nach dem Ab-<lb/>
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(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">a</hi>) = <hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>; (<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) (<hi rendition="#i">c</hi> + <hi rendition="#i">d</hi>) (<hi rendition="#i">d</hi> + <hi rendition="#i">a</hi>) = <hi rendition="#i">a c</hi> + <hi rendition="#i">b d</hi>;<lb/>
(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) = <hi rendition="#i">b</hi> + <hi rendition="#i">a c</hi>; (<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) (<hi rendition="#i">c</hi> + <hi rendition="#i">d</hi>) = <hi rendition="#i">a c</hi> + <hi rendition="#i">c b</hi> + <hi rendition="#i">b d</hi>;<lb/>
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(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">a</hi> + <hi rendition="#i">c</hi>) (<hi rendition="#i">a</hi> + <hi rendition="#i">d</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">d</hi>) (<hi rendition="#i">c</hi> + <hi rendition="#i">d</hi>) = <hi rendition="#i">a b c</hi> + <hi rendition="#i">a b d</hi> + <hi rendition="#i">a c d</hi> + <hi rendition="#i">b c d</hi>;<lb/>
(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) (<hi rendition="#i">c</hi> + <hi rendition="#i">d</hi>) (<hi rendition="#i">d</hi> + <hi rendition="#i">e</hi>) (<hi rendition="#i">e</hi> + <hi rendition="#i">a</hi>) = <hi rendition="#i">a d b</hi> + <hi rendition="#i">b e c</hi> + <hi rendition="#i">c a d</hi> + <hi rendition="#i">d b e</hi> + <hi rendition="#i">e c a</hi>;<lb/>
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(<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="#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="#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="#i">a b</hi> + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>;<lb/>
(<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi><hi rendition="#sub">1</hi>) = <hi rendition="#i">a b</hi>, (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">x</hi> + <hi rendition="#i">b</hi>) (<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>) = <hi rendition="#i">a b</hi>; <hi rendition="#i">a</hi> (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> = 1;<lb/>
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{<hi rendition="#i">a b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi> + (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi>) <hi rendition="#i">b</hi>} {<hi rendition="#i">a b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>)} = <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi>; <hi rendition="#i">a b</hi> + <hi rendition="#i">a b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi> = <hi rendition="#i">a</hi> (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>);<lb/><hi rendition="#i">a</hi> (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi>) (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b c</hi>) = <hi rendition="#i">a c</hi> (<hi rendition="#i">a</hi><hi rendition="#sub">1</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="#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">c</hi>) <hi rendition="#i">b</hi> (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi>) = 0;<lb/><hi rendition="#i">a</hi><hi rendition="#sub">1</hi> (<hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">c a</hi>) (<hi rendition="#i">c</hi> + <hi rendition="#i">a b</hi>) = 0; (<hi rendition="#i">x</hi> + <hi rendition="#i">y</hi>) (<hi rendition="#i">x</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">y z</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">x z</hi>) = 0;<lb/><hi rendition="#i">x y</hi> (<hi rendition="#i">a</hi> + <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">b</hi> + <hi rendition="#i">y</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">x</hi><hi rendition="#sub">1</hi>) = 0; <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">a b</hi> + <hi rendition="#i">a c</hi> + <hi rendition="#i">b c</hi> = 1;<lb/>
(<hi rendition="#i">x</hi> + <hi rendition="#i">y</hi>) (<hi rendition="#i">x</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">z</hi><hi rendition="#sub">1</hi>) (<hi rendition="#i">y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">z</hi>) (<hi rendition="#i">x</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">y</hi> + <hi rendition="#i">z</hi>) (<hi rendition="#i">x</hi> + <hi rendition="#i">y</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">z</hi><hi rendition="#sub">1</hi>) = 0; <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</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="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi>;<lb/>
(<hi rendition="#i">a</hi> + <hi rendition="#i">b</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="#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="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</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>);<lb/><hi rendition="#et"><hi rendition="#i">a b</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>) = 0 = <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> <hi rendition="#i">b</hi><hi rendition="#sub">1</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><lb/>
(<hi rendition="#i">b c</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi>) (<hi rendition="#i">c a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> <hi rendition="#i">a</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>) = 0; (<hi rendition="#i">b c</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi>) + (<hi rendition="#i">c a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> <hi rendition="#i">a</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>) =<lb/>
= (<hi rendition="#i">c a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> <hi rendition="#i">a</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="#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="#i">b c</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi>) = (<hi rendition="#i">b c</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> <hi rendition="#i">c</hi>) + (<hi rendition="#i">c a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi> <hi rendition="#i">a</hi>) =<lb/>
= (<hi rendition="#i">a</hi> + <hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) (<hi rendition="#i">a</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi>) = <hi rendition="#i">a</hi> (<hi rendition="#i">b</hi><hi rendition="#sub">1</hi> + <hi rendition="#i">c</hi><hi rendition="#sub">1</hi>) + <hi rendition="#i">a</hi><hi rendition="#sub">1</hi> (<hi rendition="#i">b</hi> + <hi rendition="#i">c</hi>) = etc.<lb/></p>
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[384/0404] Neunte Vorlesung. desgleichen unsymmetrisch, aber einfacher, als: (a + b c) (b + a c) = a (b + c) + b (a + c), etc. — indem diese Ausdrücke alle durch Ausmultipliziren, nach dem Ab- sorptionsgesetze auf a b + a c + b c hinauskommen. — χ) (a + b c) b = (a + c) b; a (a b + b c) = a b; (a b + a c + b c) a b c = a b c; (b + a c) (c + d) = a c + b c + b d; a + b (c + d) + (a + b x) c = a + b (c + d); (a + b) (b + a c) = b + a c; (a + b) (b + a) = a + b; (a + b) (b + c) (c + d) (d + a) = a c + b d; (a + b) (b + c) = b + a c; (a + b) (b + c) (c + d) = a c + c b + b d; (a + b) (b + c) (c + d) (d + e) = b d + c (a d + a e + b e); (a + b) (b + c) (c + d) (d + e) (e + f) = a c d f + a c e + b c e + b d e + b d f; (a + b) (a + c) (a + d) (b + c) (b + d) (c + d) = a b c + a b d + a c d + b c d; (a + b) (b + c) (c + d) (d + e) (e + a) = a d b + b e c + c a d + d b e + e c a; (a + b + c) (a + b + d) (a + c + d) (b + c + d) = a b + a c + a d + b c + b d + c d; a (b + c) c (a + b) = a c; (a + b c) (b + a c) = a b + a c + b c = (a + b) (a b + a c + b c); (a b + c d) (a + b) (c + d) = a b (c + d) + (a + b) c d. a (b + c) + c1 = a + c1; a (b + c) (a1 + b1) c1 = 0; a1 b c (b1 + a c + a1 c1) = 0; (a + b) (a1 + b1) = a b1 + a1 b, (a + b1) (a1 + b) = a b + a1 b1; (a + b1 c) (b + a1 c1) = a b, (a1 x + b) (a + b1 x1) = a b; a (b + c) + a1 + b1 c1 = 1; (a + b1 c1) (a c + b1) = a c + b1 c1; a (b1 + c d) b (c1 + d) = a b c d; (a b1 c + a1 b c1) (a b c1 + a1 b1 c) = 0; (a1 + b c) (a + b1 c1) = a b c + a1 b1 c1; (a1 + b1) (a b + a c + b c) = c (a b1 + a1 b); (a1 + b1) a (b c1 + b1 c) = a b1 c; a (b + c) (c1 + a b + a1 b1) = a b; a + b1 + c1 + b (a c1 + a1 c) = 1; {a b1 c + (a1 + c1) b} {a b1 c1 + a1 (b + c)} = a1 b; a b + a b1 c = a (b + c); a (a1 + b1 c) (a1 + b c) = a c (a1 + b1) (a1 + b) = a (b1 + a1 c) b (a1 + c1) = 0; a1 (b1 + c1) (b + c a) (c + a b) = 0; (x + y) (x1 + y z1) (y1 + x z) = 0; x y (a + x1) (a1 + y1) (b + y1) (b1 + x1) = 0; a1 + b1 + c1 + a b + a c + b c = 1; (x + y) (x1 + z1) (y1 + z) (x1 + y + z) (x + y1 + z1) = 0; a1 + b1 + c (a b1 + a1 b) = a1 + b1; (a + b) (a b1 + a1 b) = a b1 + a1 b = (a1 + b1) (a b1 + a1 b); a b (a b1 + a1 b) = 0 = a1 b1 (a b1 + a1 b); (b c1 + b1 c) (c a1 + c1 a) (a b1 + a1 b) = 0; (b c1 + b1 c) + (c a1 + c1 a) + (a b1 + a1 b) = = (c a1 + c1 a) + (a b1 + a1 b) = (a b1 + a1 b) + (b c1 + b1 c) = (b c1 + b1 c) + (c a1 + c1 a) = = (a + b + c) (a1 + b1 + c1) = a (b1 + c1) + a1 (b + c) = etc.

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Zitationshilfe:	Schröder, Ernst: Vorlesungen über die Algebra der Logik. Bd. 1. Leipzig, 1890, S. 384. In: Deutsches Textarchiv <https://www.deutschestextarchiv.de/schroeder_logik01_1890/404>, abgerufen am 09.05.2024.

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