\title	{Propiedades algebraicas}
\author	{XIAO Gang}
\email	{xiao@unice.fr}
\format	{html}

\integer{n=random(2..9)}
\text{and={\rm\ and\ }}
\text{imply=\Rightarrow}

\text{good=shuffle(
 A + B = B + A,
 A + (B + C) = (A + B) + C,
 A + (B + C) = (A + C) + B,
 A + B = A + C \imply B = C,
 A + C = B + C \imply A = B,
 (a+b)A = aA + bA,
 a(A + B) = aA + aB,
 a*b(A) = a(bA),
 a*b(A) = b(aA),
 a(bA) = b(aA),
 a(AB) = A(aB),
 (aA)B = a(AB),
 a \ne 0 \and aA = aB \imply A = B,
 A \ne 0 \and aA = bA \imply a = b,
 A \ne 0 \and aA = 0 \imply a = 0,
 a \ne 0 \and aA = 0 \imply A = 0,
 a \ne 0 \and A \ne B \imply aA \ne aB,
 A \ne 0 \and a \ne b \imply aA \ne bA,
 a \ne 0 \and A \ne 0 \imply aA \ne 0,
 A(BC) = (AB)C,
 A(B+C) = AB + AC,
 (A+B)C = AC + BC,
 AB + AC = A(B+C),
 AC + BC = (A+B)C,
 A^2B = A(AB),
 AB^2 = (AB)B
)}
\text{bad=shuffle(
 a(AB) = (aB)A,
 (aA)B = (aB)A,
 (aA)B = B(aA),
 AB = BA,
 A(BC) = (AC)B,
 A \ne 0 \and AB = AC \imply B = C,
 C \ne 0 \and AC = BC \imply A = B,
 B \ne 0 \and AB = 0 \imply A = 0,
 A \ne 0 \and AB = 0 \imply B = 0,
 A \ne 0 \and B \ne C \imply AB \ne AC,
 C \ne 0 \and A \ne B \imply AC \ne BC,
 A \ne 0 \and B \ne 0 \imply AB \ne 0,
 A(B+C) = AB + CA,
 (A+B)C = CA + BC,
 BA + AC = A(B+C),
 AC + CB = (A+B)C,
 ABA = A^2*B,
 BAB = A*B^2,
 (A-B)^2 = A^2 - B^2,
 (A+B)^2 = A^2 + 2AB + B^2,
 (A-B)^2 = A^2 - 2AB + B^2
)}

\integer{badcnt=random(1,1,2,2,3)}
\integer{goodcnt=4-\badcnt}
\text{bads=item(1..\badcnt,1,1,1,1,1,1)}
\text{gb=item(1..4,\bads,0,0,0,0,0,0)}
\text{bads=item(1..\badcnt,\bad)}
\text{goods=item(1..4-\badcnt,\good)}
\text{order=shuffle(4)}
\text{list=item(\order,\bads,\goods)}
\text{gb=item(\order,\gb)}
\text{bad=wims(positionof item 1 in \gb)}
\text{good=wims(positionof item 0 in \gb)}
\text{l1=item(1,\list)}
\text{l2=item(2,\list)}
\text{l3=item(3,\list)}
\text{l4=item(4,\list)}
\text{space=<td>   }
\if{random(1,2) = 1}{
 \text{ask=no siempre verdaderas}
 \text{cond=malas}
 \text{ans=\bad}
}{
 \text{ask=siempre verdaderas}
 \text{cond=buenas}
 \text{ans=\good}
}

\statement{Sean \(A,B,C) tres matrices \n×\n  y
\(a,b) dos números. Entre las siguientes propiedades, 
¿Cuáles de ellas son \ask?
<p><center><table border=0>
 <tr><td>1.\space<td>\(\l1)
 <tr><td>2.\space<td>\(\l2)
 <tr><td>3.\space<td>\(\l3)
 <tr><td>4.\space<td>\(\l4)
</table></center> <p>
}

\answer{Las propiedades \cond son}{\ans}{type=chset}