91Ó°ÊÓ

Prove each, where \(A, B,\) and \(C\) are arbitrary languages over \(\Sigma\) and \(x \in \Sigma\) . $$A(B \cap C) \subseteq A B \cap A C$$

A number in ALGOL (excluding the exponential form) is defined as follows: $$\langle\text { number }\rangle :=\langle\text { decimal number }\rangle :\langle\text { sign }\rangle\langle\text { decimal number }\rangle$$ \(\langle\text { decimal number }\rangle : :=\langle\text { unsigned integer }\rangle \langle\text { unsigned integer }\rangle |\) $$\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle$$ $$\langle\text { unsigned integer }\rangle : :=\langle\text { digit }\rangle :\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\operatorname{sign}\rangle : :=+|-$$ Use this grammar to answer Exercises \(60-67\). Draw a derivation tree for each ALGOL number. $$-3.76$$

Prove each, where \(A, B,\) and \(C\) are arbitrary languages over \(\Sigma\) and \(x \in \Sigma\) . $$(B \cup C) A=B A \cup C A$$

A number in ALGOL (excluding the exponential form) is defined as follows: $$\langle\text { number }\rangle :=\langle\text { decimal number }\rangle :\langle\text { sign }\rangle\langle\text { decimal number }\rangle$$ \(\langle\text { decimal number }\rangle : :=\langle\text { unsigned integer }\rangle \langle\text { unsigned integer }\rangle |\) $$\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle$$ $$\langle\text { unsigned integer }\rangle : :=\langle\text { digit }\rangle :\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\operatorname{sign}\rangle : :=+|-$$ Use this grammar to answer Exercises \(60-67\). Draw a derivation tree for each ALGOL number. $$+376$$

Prove each, where \(A, B,\) and \(C\) are arbitrary languages over \(\Sigma\) and \(x \in \Sigma\) . $$(B \cap C) A \subseteq B A \cap C A$$

A number in ALGOL (excluding the exponential form) is defined as follows: $$\langle\text { number }\rangle :=\langle\text { decimal number }\rangle :\langle\text { sign }\rangle\langle\text { decimal number }\rangle$$ \(\langle\text { decimal number }\rangle : :=\langle\text { unsigned integer }\rangle \langle\text { unsigned integer }\rangle |\) $$\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle$$ $$\langle\text { unsigned integer }\rangle : :=\langle\text { digit }\rangle :\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\operatorname{sign}\rangle : :=+|-$$ Use this grammar to answer Exercises \(60-67\). Draw a derivation tree for each ALGOL number. $$.376$$

A number in ALGOL (excluding the exponential form) is defined as follows: $$\langle\text { number }\rangle :=\langle\text { decimal number }\rangle :\langle\text { sign }\rangle\langle\text { decimal number }\rangle$$ \(\langle\text { decimal number }\rangle : :=\langle\text { unsigned integer }\rangle \langle\text { unsigned integer }\rangle |\) $$\langle\text {unsigned integer}\rangle. \langle\text {unsigned integer}\rangle$$ $$\langle\text { unsigned integer }\rangle : :=\langle\text { digit }\rangle :\langle\text { unsigned integer }\rangle\langle\text { digit }\rangle$$ $$\langle\text { digit }\rangle : := 0|1| 2|3| 4|5| 6|7| 8 | 9$$ $$\langle\operatorname{sign}\rangle : :=+|-$$ Use this grammar to answer Exercises \(60-67\). Draw a derivation tree for each ALGOL number. $$0.23$$

Prove each, where \(A, B,\) and \(C\) are arbitrary languages over \(\Sigma\) and \(x \in \Sigma\) . $$\left(A^{*} \cup B^{*}\right)^{*}=(A \cup B)^{*}$$

Determine if each is a legal expression. $$a+b *(c / d)$$

For Exercises \(68-73,\) use the following definition of a simple algebraic expression: $$\langle\text {expression}\rangle : :=\langle\text { term }\rangle |\langle\text { sign }\rangle\langle\text { term }\rangle |$$ $$\langle\text { expression }\rangle\langle\text { adding operator }\rangle\langle\text { term }\rangle$$ $$\langle\operatorname{sign}\rangle \therefore=+ 1-$$ $$\langle\text { adding operator}\rangle: :=+1-$$ $$\langle\text { term }\rangle : :=\langle\text { factor }\rangle |$$ $$\langle\text { term }\rangle\langle\text { multiplying operator }\rangle\langle\text { factor }\rangle$$ $$\langle\text { multiplying operator }\rangle := *| /$$ $$\langle\text { factor }\rangle : :=\langle\text { letter }|\rangle (\langle\text { expression }\rangle |\langle\text { expression }\rangle$$ $$\langle\text { letter }\rangle : := a|b| c | \ldots : z$$ Determine if each is a legal expression. $$a+b *(c / d)$$