PROPERTIES: 1. Associativity: 1.1. R + (S + T) = (R + S) + T 1.2. R (S T) = (R S) T 2. Commutativity: R + S = S + R 3. Distributivity: 3.1. R (S + T) = R S + R T 3.2. (S + T) R = S R + T R 4. Identity element: 4.1 R + = + R = R 4.2 R = R = R 5. Neutral element: R = R = 6. Idempotence: 6.1. R+R=R 6.2. (R^{})^{} = R^{} 7. Properties of the Kleene lock: 7.1. ^{}= 7.2. ^{} = 7.3. R^{+} = R R^{} = R^{} R 7.4. R^{} R^{} = R^{} 7.5. R^{} = + R^{+} = R^{+} + 7.6. R? = R + = + R 7.7. (R + S)^{} = (R^{} S^{} )^{} = (R^{} S)^{} R^{} 7.8. (R S)^{} = + R (S R)^{} S 7.10. R (S R)^{} = (R S)^{} R 8. Conditional properties: 8.1. If L(R) L(S), then R + S = S 8.2. If L(R^{}) L(S^{}), then R^{} S^{} = S^{} 8.3. If L(R^{}) L(S^{}), then (R + S)^{} = S^{}

1. EXAMPLE EXERCISE: a)

b)

1. legible handwriting please 2. do it following the example a) the diagrama to do with circles b) and the properties usted put them aside

EXERCISE TO SOLVE:

R=a+a(b+aa)(baa)b+a(aa+b)S=a(aa+b) DERIVATIN: R=a+a(b+aa)(baa)b+a(aa+b)=a+a(b+aa)(b+aa)+a(aa+b)=a(+(b+aa)(b+aa))+a(aa+b)=a(+(b+aa)+)+a(aa+b)=a(b+aa)+a(aa+b)=a(aa+b)+a(aa+b)=a(aa+b) important Information: EXAMPLE descrebe it formacli CONSTDER THE AUTOMATON A=(Q,,,90,F) DFFIUIDE AS 1={0,1},Q={90,q1,q2},F={q2} AND TEE TRGU SIILONS GIUEN BY: \begin{tabular}{lc|cc} (q0,0)={q0,q1} & & 0 & 1 \\ (q0,1)={q1} & q0 & {q0,q1} & {q0} \\ (q1,1)={q2} & q1 & & {q2} \\ & q2 & & \end{tabular} CS Escaneado con CamScannet TRANSITLON DLAGRAM: II IS A DIRECTED GRAPH 6=(U,E) DEFINIDED AS: 1. EACh sTATE qEQ is asJoctated to a NOdE rEE 2. FOR a,p EQ and a E III(a,a)=P THeU THERE IS A TRANSITIOU CP,q E E' FROM IHE NODE ASSOCLATED WITH 9 TO THAT ASSOCIATED WIHH P. 3. THE INICTAL NODE qO has an arrow WITH NC PRECEDLNG NODES. 4. THE TERMINAL NODES QEF arE marked WIIHA R=01++0103^ S=01(1+(0+)) a) SHOW THAI THEY ARE EQUIUACENT, DEUECOOING THE EXPRESEONS (PONNT OOT THE PROOERTIES USED). A WITH E TRANSITLOUS THAT ACCEPTS THE CANGUAGE. DRAN W W HH HHHHHHHHHHHOHHHHHHH+IHHIHH W