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 matches the number regular expression by the line a number is an integer exponent.
1 is a number since it is an integer and an integer is a digit.
E-4 is an exponent , and a real can be a integer exponent.
1E-4 is a real, which is 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 matches the number regular expression by the line a number is an integer exponent.
1 is a number since it is an integer and an integer is a digit.
E-4 is an exponent , and a real can be a integer exponent.
1E-4 is a real, which is a number.
number -> integer|real
integer -> digit digit*
digit -> 0|1|2|3|4|5|6|7|8|9
exponent -> (e|E)(+|-|epsilon)integer
1*10^-4
1.) 1 is an integer
integer * 10^-4
2.) 10 is an integer
integer * integer ^-4
3.) ^-4 is an exponent
integer * integer (E)(-)integer
4.) integers are digit digit*
digit digit* digit digit* digit digit*
5.) simplify
digit digit*
6.) digit digit* is an integer thus 1E-4 is a number by definition
number -> integer|real
integer -> digit digit*
digit -> 0|1|2|3|4|5|6|7|8|9
exponent -> (e|E)(+|-|epsilon)integer
1*10^-4
1.) 1 is an integer
integer * 10^-4
2.) 10 is an integer
integer * integer ^-4
3.) ^-4 is an exponent
integer * integer (E)(-)integer
4.) integers are digit digit*
digit digit* digit digit* digit digit*
5.) simplify
digit digit*
6.) digit digit* is an integer thus 1E-4 is a number by definition
1E-4 matches the number regular expression because an integer can be a number and an integer is one or more digits. 1 is a digit. A number can also be a real number and a real number can be an integer exponent. 4 is an integer since it is a digit and can be an exponent since it can be a '-' and following a 4, which is an integer proven previously. Therefore, 1E-4 is a number.
1E-4 matches the number regular expression because an integer can be a number and an integer is one or more digits. 1 is a digit. A number can also be a real number and a real number can be an integer exponent. 4 is an integer since it is a digit and can be an exponent since it can be a '-' and following a 4, which is an integer proven previously. Therefore, 1E-4 is a number.
• Expand out the definition of integer from the previous slide as a single regular expression.
• Show that 1E-4 is a number.
Expand integer
• integer -> digit digit* • digit -> 0|1|2|3|4|5|6|7|8|9 • Therefore integer expanded would be • integer -> (0|1|2|3|4|5|6|7|8|9) (0|1|2|3|4|5|6|7|8|9)*
Show that 1 E-4 is a number
• A number per its regex is either an integer or a real • An integer as shown above is one digit concatenated with zero or more other digits ○ The character '1' is a digit ○ The character 'E' is not a digit therefore 1E-4 is not an integer • A real is an integer concatenated with an exponent or a decimal concatenated with an exponent or the empty string ○ An integer as shown above is one digit concatenated with zero or more other digits ○ An Exponent is the character 'e' or the character 'E' concatenated with the '+' or '-' or empty string concatenated with an integer § the character '1' is an integer and it is concatenated with the string "E-4" which is an exponentTherefore 1E-4 is a real and is therefore a number
• Expand out the definition of integer from the previous slide as a single regular expression.
• Show that 1E-4 is a number.
Expand integer
• integer -> digit digit*
• digit -> 0|1|2|3|4|5|6|7|8|9
• Therefore integer expanded would be
• integer -> (0|1|2|3|4|5|6|7|8|9) (0|1|2|3|4|5|6|7|8|9)*
Show that 1 E-4 is a number
• A number per its regex is either an integer or a real
• An integer as shown above is one digit concatenated with zero or more other digits
○ The character '1' is a digit
○ The character 'E' is not a digit therefore 1E-4 is not an integer
• A real is an integer concatenated with an exponent or a decimal concatenated with an exponent or the empty string
○ An integer as shown above is one digit concatenated with zero or more other digits
○ An Exponent is the character 'e' or the character 'E' concatenated with the '+' or '-' or empty string concatenated with an integer
§ the character '1' is an integer and it is concatenated with the string "E-4" which is an exponent
Therefore 1E-4 is a real and is therefore a number
((resource:20210901_184705.jpg|Resource Description for 20210901_184705.jpg))
integer -> (0|1|2|3|4|5|6|7|8|9)
a number -> a real.
a real -> an integer exponent.
1 is an integer.
E-4 is an exponent.
So 1E-4 is a real, and 1E-4 is a number
(Edited: 2021-09-01) integer -> (0|1|2|3|4|5|6|7|8|9)
a number -> a real.
a real -> an integer exponent.
1 is an integer.
E-4 is an exponent.
So 1E-4 is a real, and 1E-4 is a number
integer -> the concatenation (*) of one or more (0|1|2|3|4|5|6|7|8|9)
A real is an integer followed by an exponent.
1 is an integer followed by an exponent E-4.
A number is an integer or real
A real number is real real
1E-4 is a real number
integer -> the concatenation (*) of one or more (0|1|2|3|4|5|6|7|8|9)
A real is an integer followed by an exponent.
1 is an integer followed by an exponent E-4.
A number is an integer or real
A real number is real real
1E-4 is a real number
integer->digit digit*
digit->0|1|2|3|4|5|6|7|8|9|
number->real->integer exponent->digit digit* exponent->1 digit* exponent->1 exponent->1 exponent->1 E - integer->1 E - digit digit*->1 E - 4 digit* ->1 E - 4
<nowiki>
integer->digit digit*
digit->0|1|2|3|4|5|6|7|8|9|
number->real->integer exponent->digit digit* exponent->1 digit* exponent->1 exponent->1 exponent->1 E - integer->1 E - digit digit*->1 E - 4 digit* ->1 E - 4
</nowiki>
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:
number -> real
real -> integer exponent
integer = 1
exponent = E integer
integer = 4
1E-4 -> number
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:
number -> real
real -> integer exponent
integer = 1
exponent = E integer
integer = 4
1E-4 -> number
Single Regular Expression
integer -> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)*
Prove 1E-4 is a number.
A real can be made from an integer and an exponent put together. An exponent can be a 'E' or a 'e' followed by either '-', '+', or 'ε' followed by an integer. E-4 is an exponent since there is an 'E' followed by '-' which is then followed by a 4 which is a digit and an integer. 1 is an integer since it is a digit and E-4 is an exponent. A real is an integer followed by an exponent. Therefore 1E-4 is a number since 1E-4 is real.
Rules used: number -> real real -> integer exponent integer -> digit digit* exponent→(e|E)(+|−|ε)integer digit -> 0|1|2|3|4|5|6|7|8|9
Single Regular Expression
integer -> (0|1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)*
Prove 1E-4 is a number.
A real can be made from an integer and an exponent put together.
An exponent can be a 'E' or a 'e' followed by either '-', '+', or 'ε' followed by an integer.
E-4 is an exponent since there is an 'E' followed by '-' which is then followed by a 4 which is a digit and an integer.
1 is an integer since it is a digit and E-4 is an exponent. A real is an integer followed by an exponent.
Therefore 1E-4 is a number since 1E-4 is real.
Rules used:
number -> real
real -> integer exponent
integer -> digit digit*
exponent→(e|E)(+|−|ε)integer
digit -> 0|1|2|3|4|5|6|7|8|9
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