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-- Sep 1 In-Class Exercise

integer as a single regular expression

Integer --> (0|1|2|3|4|5|6|7|8|9) (0|1|2|3|4|5|6|7|8|9) ⋆

Show that 1E-4 is a number

 1 is a digit
 E-4 is an exponent 
 a digit can be an integer
 integer exponent is a real
 integer real is a number

(Edited: 2021-09-01)

integer as a single regular expression Integer --> (0|1|2|3|4|5|6|7|8|9) (0|1|2|3|4|5|6|7|8|9) ⋆ Show that 1E-4 is a number 1 is a digit E-4 is an exponent a digit can be an integer integer exponent is a real integer real is a number

-- Sep 1 In-Class Exercise

 1. Integer -> 0|1|2|3|4|5|6|7|8|9
 
 2. 1E-4
 1 -> number -> integer|real -> 
 digit digit* -> 0|1|2|3... therefore 1 is a number.

 E -> number -> integer|real -> integer exponent|decimal(exponent|epsilon)
  -> (e|E)(+|-|epsilon)integer therefore E is a number.

 - -> number -> integer|real -> integer exponent|decimal(exponent|epsilon)
 -> (e|E)(+|-|epsilon)integer therefore - is a number.

 4 -> number -> integer|real -> digit digit*
 -> 0|1|2|3... therefore 4 is a number.

(Edited: 2021-09-01)

1. Integer -> 0|1|2|3|4|5|6|7|8|9 2. 1E-4 1 -> number -> integer|real -> digit digit* -> 0|1|2|3... '''therefore 1 is a number.''' E -> number -> integer|real -> integer exponent|decimal(exponent|epsilon) -> (e|E)(+|-|epsilon)integer '''therefore E is a number.''' - -> number -> integer|real -> integer exponent|decimal(exponent|epsilon) -> (e|E)(+|-|epsilon)integer '''therefore - is a number.''' 4 -> number -> integer|real -> digit digit* -> 0|1|2|3... '''therefore 4 is a number.'''

-- Sep 1 In-Class Exercise

Integer --> (0|1|2|3|4|5|6|7|8|9) (0|1|2|3|4|5|6|7|8|9)*

1E-4
By the first line, number can be real. By the third line, real is integer followed by exponent.
(1) 1 is an integer since 1 is a digit and an integer can be one of more digit.
(2) Exponent is defined as E followed by (+|-|e) => E-4 is an exponent
From (1) and (2) => 1E-4 is a real, thereby, a number.

(Edited: 2021-09-01)

Integer --> (0|1|2|3|4|5|6|7|8|9) (0|1|2|3|4|5|6|7|8|9)* 1E-4 By the first line, number can be real. By the third line, real is integer followed by exponent. (1) 1 is an integer since 1 is a digit and an integer can be one of more digit. (2) Exponent is defined as E followed by (+|-|e) => E-4 is an exponent From (1) and (2) => 1E-4 is a real, thereby, a number.

-- Sep 1 In-Class Exercise

Regular expression 1E-4 integer→(0|1|2|3|4|5|6|7|8|9

An integer is number and E-4 is exponent is number

so 1E-4 is number

Regular expression 1E-4 integer→(0|1|2|3|4|5|6|7|8|9 An integer is number and E-4 is exponent is number so 1E-4 is number

-- Sep 1 In-Class Exercise

Integer --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)*

1E-4 is a number because 1 is an integer that is followed by the exponent E-4.

Integer --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)* 1E-4 is a number because 1 is an integer that is followed by the exponent E-4.

-- Sep 1 In-Class Exercise

integer --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ number --> real --> integer exponent --> digit digit⋆ (e|E)(+|-|ε)integer --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ (e|E)(+|-|ε) digit digit⋆ --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ (e|E)(+|-|ε) (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ 1E-4: (0)(1)^1 (E)(-)(0)(4)^1

<nowiki>integer --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ number --> real --> integer exponent --> digit digit⋆ (e|E)(+|-|ε)integer --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ (e|E)(+|-|ε) digit digit⋆ --> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ (e|E)(+|-|ε) (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)⋆ 1E-4: (0)(1)^1 (E)(-)(0)(4)^1</nowiki>

-- Sep 1 In-Class Exercise

 Expand integer:
 integer → (0∣∣1∣∣2∣∣3∣∣4∣∣5∣∣6∣∣7∣∣8∣∣9) (0∣∣1∣∣2∣∣3∣∣4∣∣5∣∣6∣∣7∣∣8∣∣9)⋆
 
 Show that 1E-4 is a number: 
 1 is a digit - digit → 0∣∣1	∣∣2∣∣3∣∣4∣∣5∣∣6∣∣7∣∣8∣∣9 
 1 is an integer - integer → digit	 digit⋆ 
 4 is a digit - digit → 0∣∣1∣∣2∣∣3∣∣4	∣∣5∣∣6∣∣7∣∣8∣∣9 
 4 is an integer - integer → digit	 digit⋆ 
 E-4 is an exponent - exponent → (e∣∣E	)(+∣∣−	∣∣ε)integer	
 1E-4 is a real - real → integer	 exponent	∣∣decimal(exponent∣∣ε)
 1E-4 is a number - number → integer∣∣real

(Edited: 2021-09-01)

Expand integer: integer → (0∣∣1∣∣2∣∣3∣∣4∣∣5∣∣6∣∣7∣∣8∣∣9) (0∣∣1∣∣2∣∣3∣∣4∣∣5∣∣6∣∣7∣∣8∣∣9)⋆ Show that 1E-4 is a number: 1 is a digit - digit → 0∣∣'''1'''∣∣2∣∣3∣∣4∣∣5∣∣6∣∣7∣∣8∣∣9 1 is an integer - integer → '''digit''' digit⋆ 4 is a digit - digit → 0∣∣1∣∣2∣∣3∣∣'''4'''∣∣5∣∣6∣∣7∣∣8∣∣9 4 is an integer - integer → '''digit''' digit⋆ E-4 is an exponent - exponent → (e∣∣'''E''')(+∣∣'''−'''∣∣ε)'''integer''' 1E-4 is a real - real → '''integer''' '''exponent'''∣∣decimal(exponent∣∣ε) 1E-4 is a number - number → integer∣∣'''real'''

-- Sep 1 In-Class Exercise

Integer → (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9) * 1E-4 number is real, real is integer exponent so real → (0|1|2|3|4|5|6|7|8|9) * (e|E) (+|-|e) (0|1|2|3|4|5|6|7|8|9) * Based on these rules 1 fits integer and E-4 fits exponent.

<nowiki>Integer → (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9) * 1E-4 number is real, real is integer exponent so real → (0|1|2|3|4|5|6|7|8|9) * (e|E) (+|-|e) (0|1|2|3|4|5|6|7|8|9) * Based on these rules 1 fits integer and E-4 fits exponent.</nowiki>

-- Sep 1 In-Class Exercise

 integer —> digit digit *
 digit —> 0|1|2|3|4|5|6|7|8|9

 Number is real. Therefore real is an integer with an exponent. 
 An exponent is either e or E with either +,-,or empty string
  along with an integer. 
 An integer is a digit digit *. A digit is either 0,1,2,3,4,5,6,
 7, 8,9.

(Edited: 2021-09-01)

integer —> digit digit * digit —> 0|1|2|3|4|5|6|7|8|9 Number is real. Therefore real is an integer with an exponent. An exponent is either e or E with either +,-,or empty string along with an integer. An integer is a digit digit *. A digit is either 0,1,2,3,4,5,6, 7, 8,9.

-- Sep 1 In-Class Exercise

IE-4

integer -> (0|1|2|3|4|5|6|7|8|9)*

number -> integer|real

1 - digit

E - exponent is real

- - concatenated to Exponent which is a real

4 - digit

A number contains integer|real. Since 1 is a digit, it is real which makes it a number. 1 is concatenated with an exponent with the necessary '-' and digit, which altogether is a real. An integer concatenated with a real is a number.

(Edited: 2021-09-01)

IE-4 integer -> (0|1|2|3|4|5|6|7|8|9)* number -> integer|real 1 - digit E - exponent is real - - concatenated to Exponent which is a real 4 - digit A number contains integer|real. Since 1 is a digit, it is real which makes it a number. 1 is concatenated with an exponent with the necessary '-' and digit, which altogether is a real. An integer concatenated with a real is a number.