He . means concatenation
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^{}
SOLVE:
Select the number in those regular expressions that are equivalent: 1) R=(1+0+)01S=1?01 2)
R=110?0S=1+(0+) 3) R=1+S=11 4) R=1+S= 5) R=0+1S=00(1+).
