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Questions and Answers
Which of the following regular expressions describes the language of all strings over {a, b} that contain at least one 'a'?
Which of the following regular expressions describes the language of all strings over {a, b} that contain at least one 'a'?
- (a+b)*
- b*ab* (correct)
- b*ab*a*
- b* + a
Which regular expression represents the language of all strings over {0, 1} that start with '1' and end with '0'?
Which regular expression represents the language of all strings over {0, 1} that start with '1' and end with '0'?
- 0(0+1)*1
- (1+0)*
- 1(0+1)*0 (correct)
- 1*0*
Which of the following regular expressions defines the language of all strings over the alphabet {a, b} that have an even number of 'a's?
Which of the following regular expressions defines the language of all strings over the alphabet {a, b} that have an even number of 'a's?
- b*(ab*ab*)* (correct)
- (a+b)*
- a*b*
- (aa)*
Given Σ = {0, 1}, which regular expression represents the language of all strings with at least one '0' and at least one '1'?
Given Σ = {0, 1}, which regular expression represents the language of all strings with at least one '0' and at least one '1'?
Which regular expression represents the set of all strings over {a, b} that start and end with the same letter?
Which regular expression represents the set of all strings over {a, b} that start and end with the same letter?
Which regular expression describes the language of all strings over {0, 1} that contain the substring '110'?
Which regular expression describes the language of all strings over {0, 1} that contain the substring '110'?
What language is described by the regular expression (a+b)*abb?
What language is described by the regular expression (a+b)*abb?
Which regular expression represents the language of all strings over Σ = {a, b} with length divisible by 2?
Which regular expression represents the language of all strings over Σ = {a, b} with length divisible by 2?
Select the regular expression that matches strings over {0, 1} where every '1' is immediately followed by a '0'.
Select the regular expression that matches strings over {0, 1} where every '1' is immediately followed by a '0'.
Which regular expression defines the language of all strings over Σ = {0, 1} that do not contain consecutive 1s?
Which regular expression defines the language of all strings over Σ = {0, 1} that do not contain consecutive 1s?
Which identity is correct for regular expressions?
Which identity is correct for regular expressions?
Given regular expressions P and Q, which of the following is equivalent to (P+Q)R?
Given regular expressions P and Q, which of the following is equivalent to (P+Q)R?
Which of the following regular expressions generates strings that contain exactly two 'a's?
Which of the following regular expressions generates strings that contain exactly two 'a's?
Which of the following regular expressions represent all strings that contain at least one 'a' or at least one 'b'?
Which of the following regular expressions represent all strings that contain at least one 'a' or at least one 'b'?
Which regular expression represents all words containing either a double 'a' or a double 'b'?
Which regular expression represents all words containing either a double 'a' or a double 'b'?
If L = {w | w starts with b}, what is a possible regular expression for L?
If L = {w | w starts with b}, what is a possible regular expression for L?
Which regular expression correctly represents the language of all strings over {a, b} that have an odd number of a's?
Which regular expression correctly represents the language of all strings over {a, b} that have an odd number of a's?
Which of the following regular expressions describes a language that includes only the empty string?
Which of the following regular expressions describes a language that includes only the empty string?
Choose the regular expression that describes strings of odd length over the alphabet {a}.
Choose the regular expression that describes strings of odd length over the alphabet {a}.
Select the regular expression for the language where all strings start with 'a' followed by some number of 'c' and then some number of 'b'.
Select the regular expression for the language where all strings start with 'a' followed by some number of 'c' and then some number of 'b'.
Flashcards
Concatenation
Concatenation
Combining two strings, or patterns, together in a sequence.
Star/Kleene Closure
Star/Kleene Closure
A set of strings derived by repeatedly concatenating strings from a language, including the empty string.
Union (+)
Union (+)
Combining two sets of strings to include all strings from both sets.
Regular Expression (RE)
Regular Expression (RE)
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Empty Language
Empty Language
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Empty String
Empty String
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(a+b)(a+b)
(a+b)(a+b)
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(aa)*a
(aa)*a
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(aa)*
(aa)*
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(a+b)* a (a+b)*
(a+b)* a (a+b)*
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a(a+b)*
a(a+b)*
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aa+bb + (ab+ba)(aa+bb)*(ab+ba)
aa+bb + (ab+ba)(aa+bb)*(ab+ba)
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Regular Expression
Regular Expression
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Study Notes
- Operations include union, concatenation, and star/closure.
Union (+)
- R.E. (Regular Expression) of (a+b) is {a, b}.
Concatenation
- If A = aab and B = bab, then the concatenation of (A, B) is aabbab.
Regular Expression Examples
- R.E. of (1)* = {epsilon, 1, 11, 111, ...} which is an infinite language.
- R.E. of (1100)* = {epsilon, 1100, 11001100, 110011001100,...} which is an infinite language.
- R.E. of (0+1)* = {epsilon, 0, 1, 00, 01, 10, 11, 000, 001, 011, 111, 100, 101, 010, 110,...}.
- (0+1)* means any combination of 0s and 1s, including the empty string.
- (0+1) = {0,1}.
- (0+1)* = {epsilon}.
- (0+1)(0+1) = {00, 01, 10, 11}.
- (0+1)(0+1)(0+1) = {000, 001, 010, 011, 100, 101, 110, 111}.
- R.E. of ab*a = {aa, aba, abba, abbba,...}.
- R.E. examples:
- epsilon, a, aa, aaa,...
- epsilon, b, bb, bbb,...
- epsilon, a, b, ab, ba, aa, bb, ba, aaa,...}.
- The regular expression for {0, 1, 2} is (0+1+2).
- The regular expression for {epsilon, ab} is (epsilon+ab).
- The regular expression for {epsilon, 0, 00, 000,...} results into epsilon.
- The regular expression for {0, 00, 000,...} leads to 0*.
- The regular expression for {aa, ab, ba, bb} is (a+b)(a+b).
- To describe all strings of length at least 2, the regular expression is (a+b)(a+b).
- The regular expression for {aa, ab, ba, bb, aaa, aab,...} can be represented as (a+b)(a+b).
- R.E. = (a+b)*(aa+bb) creates an acceptor.
- The regular expression for {epsilon, a, b, aa, ab, ba, bb, aaa, aab,...} can be represented as (a+b)*(aa+bb).
Accepting Strings with Restrictions
- The regular expression for strings of length at most 2 is epsilon, a, b, aa, ab, ba, bb represented as R.E = (epsilon+a+b)(epsilon+a+b).
Language Examples and Regular Expressions
- L = {all words that begin with 'a' or 'c' and are followed by some number of 'b's} can be represented as (a+c)b*.
- Examples of strings include ab, cb, abb, cbb.
Regular Expression Identities
- Phi+R = R (where Phi represents the empty set).
- Phi * R = Phi
- Epsilon * R = R
- R+R = R
- RR = R*
- RR* = R*R = R+
- (R*)* = R*
- Epsilon + RR* = Epsilon* + RR.
- (P+Q)* = (PQ)* = (P*+Q*)*.
- (P+Q)R = PR + QR.
String Properties and Regular Expressions
- Language L = {w | all strings of odd length with Σ = {a}} has representation (aa)*a.
- Language L = {w | all strings of even length} has representation (aa)*.
- The language L = {w | all strings with at least one 'a'} with Σ = {a, b} represented as (a+b)a(a+b).
- Language L = {w | all strings having exactly two 'a's} can be represented as babab*.
- The language L = {w | at least one 'a' and at least one 'b'} can be represented as [(a+b)*a(a+b)b(a+b)] + [(a+b)*b(a+b)a(a+b)].
- Represents all words starting with 'a': a(a+b)*.
- Represents all words starting with 'b': b(a+b)*.
- Represents all words ending with 'a': (a+b)*a.
- Represents all words ending with 'b': (a+b)*b.
- Represents all words starting with 'a's: a+(a+b)*.
- To enforce an even number of 'a's and an even number of 'b's, the regular expression is aa + bb + (ab+ba)(aa+bb)*(ab+ba).
- For strings that start and end with the same letter, use (a+b) + (a(a+b)*a) + (b(a+b)*b).
- The regular expression for (L = {w | that have odd numbers of 'a's}) is ba(bab*)*.
- To write a regular expression for the language that accept that have even length => ((a+b) (a+b))*.
- To write a regular expression for the language that accept that have odd length => (a+b)((a+b) (a+b))*.
- For strings with double 'a' or double 'b', the regular expression is (a+b)*(aa+bb).
- For words having double 'a' or double 'b', the regular expression is (a+b)(aa+bb)(a+b).
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